2.2 Conditional Statements -...

§ 2.2 Conditional Statements

Definition

How did we define a proposition?

DefinitionA proposition is a statement that can be true or false but not both.

Conditional propositions are compound statements. We denote themas p→ q and we think “if p then q”. These are sometimes calledimplications, where

p is called the hypothesis (antecedent)

q is called the conclusion (consequent)

Definition

How did we define a proposition?

DefinitionA proposition is a statement that can be true or false but not both.

Conditional propositions are compound statements. We denote themas p→ q and we think “if p then q”. These are sometimes calledimplications, where

p is called the hypothesis (antecedent)

q is called the conclusion (consequent)

Definition

How did we define a proposition?

DefinitionA proposition is a statement that can be true or false but not both.

Conditional propositions are compound statements. We denote themas p→ q and we think “if p then q”. These are sometimes calledimplications, where

p is called the hypothesis (antecedent)

q is called the conclusion (consequent)

Operator Precedence

Operator Precedence

1 ¬2 ∧3 ∨ and ⊕ from left to right

4 →

Operator Precedence

Operator Precedence

1 ¬2 ∧3 ∨ and ⊕ from left to right4 →

Examples

ExampleLet p be the statement ‘I am hungry’ and let q be the statement ’I willeat something’. Write the following in symbols.

1 If I am hungry then I will eat something.

p→ q2 I am hungry and I will eat something.

p ∧ q3 If I am not hungry then I will not eat something.¬p→ ¬q

4 If I will eat something then I am hungry.q→ p

Examples

ExampleLet p be the statement ‘I am hungry’ and let q be the statement ’I willeat something’. Write the following in symbols.

1 If I am hungry then I will eat something.p→ q

2 I am hungry and I will eat something.p ∧ q

3 If I am not hungry then I will not eat something.¬p→ ¬q

4 If I will eat something then I am hungry.q→ p

Examples

ExampleLet p be the statement ‘I am hungry’ and let q be the statement ’I willeat something’. Write the following in symbols.

1 If I am hungry then I will eat something.p→ q

2 I am hungry and I will eat something.

p ∧ q3 If I am not hungry then I will not eat something.¬p→ ¬q

4 If I will eat something then I am hungry.q→ p

Examples

ExampleLet p be the statement ‘I am hungry’ and let q be the statement ’I willeat something’. Write the following in symbols.

1 If I am hungry then I will eat something.p→ q

2 I am hungry and I will eat something.p ∧ q

3 If I am not hungry then I will not eat something.¬p→ ¬q

4 If I will eat something then I am hungry.q→ p

Examples

ExampleLet p be the statement ‘I am hungry’ and let q be the statement ’I willeat something’. Write the following in symbols.

1 If I am hungry then I will eat something.p→ q

2 I am hungry and I will eat something.p ∧ q

3 If I am not hungry then I will not eat something.

¬p→ ¬q4 If I will eat something then I am hungry.

q→ p

Examples

ExampleLet p be the statement ‘I am hungry’ and let q be the statement ’I willeat something’. Write the following in symbols.

1 If I am hungry then I will eat something.p→ q

2 I am hungry and I will eat something.p ∧ q

3 If I am not hungry then I will not eat something.¬p→ ¬q

4 If I will eat something then I am hungry.q→ p

Examples

ExampleLet p be the statement ‘I am hungry’ and let q be the statement ’I willeat something’. Write the following in symbols.

1 If I am hungry then I will eat something.p→ q

2 I am hungry and I will eat something.p ∧ q

3 If I am not hungry then I will not eat something.¬p→ ¬q

4 If I will eat something then I am hungry.

q→ p

Examples

ExampleLet p be the statement ‘I am hungry’ and let q be the statement ’I willeat something’. Write the following in symbols.

1 If I am hungry then I will eat something.p→ q

2 I am hungry and I will eat something.p ∧ q

3 If I am not hungry then I will not eat something.¬p→ ¬q

4 If I will eat something then I am hungry.q→ p

Truth Table for Implication

p q p→ qT T TT F FF T TF F T

Reasoningthink ‘if p is true then q will be found to be true’p→ q simply tells us that we will not have p true and q false atthe same time. It does not say ‘p caused q’.If q is true then p→ q is always true. We are looking for theconclusion, and if the conclusion is true, the hypothesis isirrelevant.

The implication ‘If 2 + 2 = 1 then I am the President’ is true simplybecause p : 2 + 2 = 1 is false, so there is no scenario where p is trueand q is false.

Truth Table for Implication

p q p→ qT T TT F FF T TF F T

Reasoningthink ‘if p is true then q will be found to be true’

p→ q simply tells us that we will not have p true and q false atthe same time. It does not say ‘p caused q’.If q is true then p→ q is always true. We are looking for theconclusion, and if the conclusion is true, the hypothesis isirrelevant.

The implication ‘If 2 + 2 = 1 then I am the President’ is true simplybecause p : 2 + 2 = 1 is false, so there is no scenario where p is trueand q is false.

Truth Table for Implication

p q p→ qT T TT F FF T TF F T

Reasoningthink ‘if p is true then q will be found to be true’p→ q simply tells us that we will not have p true and q false atthe same time. It does not say ‘p caused q’.

If q is true then p→ q is always true. We are looking for theconclusion, and if the conclusion is true, the hypothesis isirrelevant.

The implication ‘If 2 + 2 = 1 then I am the President’ is true simplybecause p : 2 + 2 = 1 is false, so there is no scenario where p is trueand q is false.

Truth Table for Implication

p q p→ qT T TT F FF T TF F T

Reasoningthink ‘if p is true then q will be found to be true’p→ q simply tells us that we will not have p true and q false atthe same time. It does not say ‘p caused q’.If q is true then p→ q is always true. We are looking for theconclusion, and if the conclusion is true, the hypothesis isirrelevant.

The implication ‘If 2 + 2 = 1 then I am the President’ is true simplybecause p : 2 + 2 = 1 is false, so there is no scenario where p is trueand q is false.

Truth Table for Implication

p q p→ qT T TT F FF T TF F T

Reasoningthink ‘if p is true then q will be found to be true’p→ q simply tells us that we will not have p true and q false atthe same time. It does not say ‘p caused q’.If q is true then p→ q is always true. We are looking for theconclusion, and if the conclusion is true, the hypothesis isirrelevant.

The implication ‘If 2 + 2 = 1 then I am the President’ is true simplybecause p : 2 + 2 = 1 is false, so there is no scenario where p is trueand q is false.

Here’s An Example

Another way to think of this: let p is “I am elected” and q is “I willlower taxes”. Think of it as the only way for the politician to breakhis/her promise is to get elected and not lower taxes, so all otherstatements are true.

ExampleGiven that p is false and q is true, determine the truth of the followingpropositions:

1 (p ∨ q)→ (p⊕ q)T → T is a true statement

2 (¬p ∧ q)→ pT → F is a false statement

3 (p→ ¬q) ∨ (p→ q)T ∨ T is a true statement

Here’s An Example

Another way to think of this: let p is “I am elected” and q is “I willlower taxes”. Think of it as the only way for the politician to breakhis/her promise is to get elected and not lower taxes, so all otherstatements are true.

ExampleGiven that p is false and q is true, determine the truth of the followingpropositions:

1 (p ∨ q)→ (p⊕ q)

T → T is a true statement2 (¬p ∧ q)→ p

T → F is a false statement3 (p→ ¬q) ∨ (p→ q)

T ∨ T is a true statement

Here’s An Example

Another way to think of this: let p is “I am elected” and q is “I willlower taxes”. Think of it as the only way for the politician to breakhis/her promise is to get elected and not lower taxes, so all otherstatements are true.

ExampleGiven that p is false and q is true, determine the truth of the followingpropositions:

1 (p ∨ q)→ (p⊕ q)T → T is a true statement

2 (¬p ∧ q)→ pT → F is a false statement

3 (p→ ¬q) ∨ (p→ q)T ∨ T is a true statement

Here’s An Example

Another way to think of this: let p is “I am elected” and q is “I willlower taxes”. Think of it as the only way for the politician to breakhis/her promise is to get elected and not lower taxes, so all otherstatements are true.

ExampleGiven that p is false and q is true, determine the truth of the followingpropositions:

1 (p ∨ q)→ (p⊕ q)T → T is a true statement

2 (¬p ∧ q)→ p

T → F is a false statement3 (p→ ¬q) ∨ (p→ q)

T ∨ T is a true statement

Here’s An Example

Another way to think of this: let p is “I am elected” and q is “I willlower taxes”. Think of it as the only way for the politician to breakhis/her promise is to get elected and not lower taxes, so all otherstatements are true.

ExampleGiven that p is false and q is true, determine the truth of the followingpropositions:

1 (p ∨ q)→ (p⊕ q)T → T is a true statement

2 (¬p ∧ q)→ pT → F is a false statement

3 (p→ ¬q) ∨ (p→ q)T ∨ T is a true statement

Here’s An Example

Another way to think of this: let p is “I am elected” and q is “I willlower taxes”. Think of it as the only way for the politician to breakhis/her promise is to get elected and not lower taxes, so all otherstatements are true.

ExampleGiven that p is false and q is true, determine the truth of the followingpropositions:

1 (p ∨ q)→ (p⊕ q)T → T is a true statement

2 (¬p ∧ q)→ pT → F is a false statement

3 (p→ ¬q) ∨ (p→ q)

T ∨ T is a true statement

Here’s An Example

Another way to think of this: let p is “I am elected” and q is “I willlower taxes”. Think of it as the only way for the politician to breakhis/her promise is to get elected and not lower taxes, so all otherstatements are true.

ExampleGiven that p is false and q is true, determine the truth of the followingpropositions:

1 (p ∨ q)→ (p⊕ q)T → T is a true statement

2 (¬p ∧ q)→ pT → F is a false statement

3 (p→ ¬q) ∨ (p→ q)T ∨ T is a true statement

Ways To State An Implication

if p then q

p is sufficient for q

q if p

q when p

a necessary condition for p is q

a sufficient condition for q is p

q unless ¬p

p implies q

p only if q

q whenever p

q follows from p

Ways To State An Implication

if p then q

p is sufficient for q

q if p

q when p

a necessary condition for p is q

a sufficient condition for q is p

q unless ¬p

p implies q

p only if q

q whenever p

q follows from p

Ways To State An Implication

if p then q

p is sufficient for q

q if p

q when p

a necessary condition for p is q

a sufficient condition for q is p

q unless ¬p

p implies q

p only if q

q whenever p

q follows from p

Ways To State An Implication

if p then q

p is sufficient for q

q if p

q when p

a necessary condition for p is q

a sufficient condition for q is p

q unless ¬p

p implies q

p only if q

q whenever p

q follows from p

Ways To State An Implication

if p then q

p is sufficient for q

q if p

q when p

a necessary condition for p is q

a sufficient condition for q is p

q unless ¬p

p implies q

p only if q

q whenever p

q follows from p

Ways To State An Implication

if p then q

p is sufficient for q

q if p

q when p

a necessary condition for p is q

a sufficient condition for q is p

q unless ¬p

p implies q

p only if q

q whenever p

q follows from p

Ways To State An Implication

if p then q

p is sufficient for q

q if p

q when p

a necessary condition for p is q

a sufficient condition for q is p

q unless ¬p

p implies q

p only if q

q whenever p

q follows from p

Ways To State An Implication

if p then q

p is sufficient for q

q if p

q when p

a necessary condition for p is q

a sufficient condition for q is p

q unless ¬p

p implies q

p only if q

q whenever p

q follows from p

Ways To State An Implication

if p then q

p is sufficient for q

q if p

q when p

a necessary condition for p is q

a sufficient condition for q is p

q unless ¬p

p implies q

p only if q

q whenever p

q follows from p

Ways To State An Implication

if p then q

p is sufficient for q

q if p

q when p

a necessary condition for p is q

a sufficient condition for q is p

q unless ¬p

p implies q

p only if q

q whenever p

q follows from p

Ways To State An Implication

if p then q

p is sufficient for q

q if p

q when p

a necessary condition for p is q

a sufficient condition for q is p

q unless ¬p

p implies q

p only if q

q whenever p

q follows from p

Clarification on Ways to State Implications

There will be three we will look at.

p only if qThis says that p is only true under the condition that q is true; inother words, it cannot be the case that p is true but q is false.

a necessary condition for p is qThis says that if p is true then necessarily q is true. This is thesame as the previous statement.

a sufficient condition for q is pThis states that the truth of p is sufficient for the truth of q. Inother words, the truth of p implies the truth of q, or that p impliesq.

Clarification on Ways to State Implications

There will be three we will look at.

p only if q

This says that p is only true under the condition that q is true; inother words, it cannot be the case that p is true but q is false.

a necessary condition for p is qThis says that if p is true then necessarily q is true. This is thesame as the previous statement.

a sufficient condition for q is pThis states that the truth of p is sufficient for the truth of q. Inother words, the truth of p implies the truth of q, or that p impliesq.

Clarification on Ways to State Implications

There will be three we will look at.

p only if qThis says that p is only true under the condition that q is true; inother words, it cannot be the case that p is true but q is false.

a necessary condition for p is qThis says that if p is true then necessarily q is true. This is thesame as the previous statement.

a sufficient condition for q is pThis states that the truth of p is sufficient for the truth of q. Inother words, the truth of p implies the truth of q, or that p impliesq.

Clarification on Ways to State Implications

There will be three we will look at.

p only if qThis says that p is only true under the condition that q is true; inother words, it cannot be the case that p is true but q is false.

a necessary condition for p is q

This says that if p is true then necessarily q is true. This is thesame as the previous statement.

a sufficient condition for q is pThis states that the truth of p is sufficient for the truth of q. Inother words, the truth of p implies the truth of q, or that p impliesq.

Clarification on Ways to State Implications

There will be three we will look at.

p only if qThis says that p is only true under the condition that q is true; inother words, it cannot be the case that p is true but q is false.

a necessary condition for p is qThis says that if p is true then necessarily q is true. This is thesame as the previous statement.

a sufficient condition for q is pThis states that the truth of p is sufficient for the truth of q. Inother words, the truth of p implies the truth of q, or that p impliesq.

Clarification on Ways to State Implications

There will be three we will look at.

p only if qThis says that p is only true under the condition that q is true; inother words, it cannot be the case that p is true but q is false.

a necessary condition for p is qThis says that if p is true then necessarily q is true. This is thesame as the previous statement.

a sufficient condition for q is p

This states that the truth of p is sufficient for the truth of q. Inother words, the truth of p implies the truth of q, or that p impliesq.

Clarification on Ways to State Implications

There will be three we will look at.

p only if qThis says that p is only true under the condition that q is true; inother words, it cannot be the case that p is true but q is false.

a necessary condition for p is qThis says that if p is true then necessarily q is true. This is thesame as the previous statement.

a sufficient condition for q is pThis states that the truth of p is sufficient for the truth of q. Inother words, the truth of p implies the truth of q, or that p impliesq.

Necessary and Sufficient Conditions

The difference between the two is one guarantees the other occurswhereas the other must be true for the other to occur.

DefinitionA necessary condition for S is a condition that must be satisfied inorder for S to occur.

ExampleA necessary condition for getting an A in this class is to take the final.This means that if you do not take the final you will not get an A butyou are not guaranteed to get an A just because you take the final.

Necessary and Sufficient Conditions

The difference between the two is one guarantees the other occurswhereas the other must be true for the other to occur.

DefinitionA necessary condition for S is a condition that must be satisfied inorder for S to occur.

ExampleA necessary condition for getting an A in this class is to take the final.This means that if you do not take the final you will not get an A butyou are not guaranteed to get an A just because you take the final.

Necessary and Sufficient Conditions

The difference between the two is one guarantees the other occurswhereas the other must be true for the other to occur.

DefinitionA necessary condition for S is a condition that must be satisfied inorder for S to occur.

ExampleA necessary condition for getting an A in this class is to take the final.This means that if you do not take the final you will not get an A butyou are not guaranteed to get an A just because you take the final.

Necessary and Sufficient Conditions

DefinitionA sufficient condition for S is a condition, if satisfied, guarantees thatS occurs.

ExampleA sufficient condition for getting an A in this class is to get an A onall assignments. It is not saying that the only way to get an A in theclass is to get an A on all assignments, but but rather that it gives youa way to guarantee the desired outcome.

Necessary and Sufficient Conditions

DefinitionA sufficient condition for S is a condition, if satisfied, guarantees thatS occurs.

ExampleA sufficient condition for getting an A in this class is to get an A onall assignments. It is not saying that the only way to get an A in theclass is to get an A on all assignments, but but rather that it gives youa way to guarantee the desired outcome.

An Example

ExampleDetermine which of the following are necessary or sufficientconditions.

1 Being a mammal is a sufficient condition for being human.

False - there are non-human mammals2 Being human is a sufficient condition for being a mammal.

True - being human guarantees that one is a mammal, so if wehave a non-mammal, it cannot be human

3 Being alive is a necessary condition for having a right to life.True - nothing that is not alive can have a right to life

An Example

ExampleDetermine which of the following are necessary or sufficientconditions.

1 Being a mammal is a sufficient condition for being human.False - there are non-human mammals

2 Being human is a sufficient condition for being a mammal.True - being human guarantees that one is a mammal, so if wehave a non-mammal, it cannot be human

3 Being alive is a necessary condition for having a right to life.True - nothing that is not alive can have a right to life

An Example

ExampleDetermine which of the following are necessary or sufficientconditions.

1 Being a mammal is a sufficient condition for being human.False - there are non-human mammals

2 Being human is a sufficient condition for being a mammal.

True - being human guarantees that one is a mammal, so if wehave a non-mammal, it cannot be human

3 Being alive is a necessary condition for having a right to life.True - nothing that is not alive can have a right to life

An Example

ExampleDetermine which of the following are necessary or sufficientconditions.

1 Being a mammal is a sufficient condition for being human.False - there are non-human mammals

2 Being human is a sufficient condition for being a mammal.True - being human guarantees that one is a mammal, so if wehave a non-mammal, it cannot be human

3 Being alive is a necessary condition for having a right to life.True - nothing that is not alive can have a right to life

An Example

ExampleDetermine which of the following are necessary or sufficientconditions.

1 Being a mammal is a sufficient condition for being human.False - there are non-human mammals

2 Being human is a sufficient condition for being a mammal.True - being human guarantees that one is a mammal, so if wehave a non-mammal, it cannot be human

3 Being alive is a necessary condition for having a right to life.

True - nothing that is not alive can have a right to life

An Example

ExampleDetermine which of the following are necessary or sufficientconditions.

1 Being a mammal is a sufficient condition for being human.False - there are non-human mammals

2 Being human is a sufficient condition for being a mammal.True - being human guarantees that one is a mammal, so if wehave a non-mammal, it cannot be human

3 Being alive is a necessary condition for having a right to life.True - nothing that is not alive can have a right to life

Properties

We have some properties for these operators and we can prove themusing truth tables.

1 Commutativity p ∧ q ≡ q ∧ p

2 Associativity p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r3 Distribution p ∨ (q ∨ r) ≡ (p ∨ q) ∧ (p ∨ r)4 Idempotent p ∨ p ≡ p and p ∧ p ≡ p5 ¬(¬p) ≡ p6 DeMorgan’s Laws

1 ¬(p ∨ q) ≡ (¬p) ∧ (¬q)2 ¬(p ∧ q) ≡ (¬p) ∨ (¬q)

Properties

We have some properties for these operators and we can prove themusing truth tables.

1 Commutativity p ∧ q ≡ q ∧ p2 Associativity p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r

3 Distribution p ∨ (q ∨ r) ≡ (p ∨ q) ∧ (p ∨ r)4 Idempotent p ∨ p ≡ p and p ∧ p ≡ p5 ¬(¬p) ≡ p6 DeMorgan’s Laws

1 ¬(p ∨ q) ≡ (¬p) ∧ (¬q)2 ¬(p ∧ q) ≡ (¬p) ∨ (¬q)

Properties

We have some properties for these operators and we can prove themusing truth tables.

1 Commutativity p ∧ q ≡ q ∧ p2 Associativity p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r3 Distribution p ∨ (q ∨ r) ≡ (p ∨ q) ∧ (p ∨ r)

4 Idempotent p ∨ p ≡ p and p ∧ p ≡ p5 ¬(¬p) ≡ p6 DeMorgan’s Laws

1 ¬(p ∨ q) ≡ (¬p) ∧ (¬q)2 ¬(p ∧ q) ≡ (¬p) ∨ (¬q)

Properties

We have some properties for these operators and we can prove themusing truth tables.

1 Commutativity p ∧ q ≡ q ∧ p2 Associativity p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r3 Distribution p ∨ (q ∨ r) ≡ (p ∨ q) ∧ (p ∨ r)4 Idempotent p ∨ p ≡ p and p ∧ p ≡ p

5 ¬(¬p) ≡ p6 DeMorgan’s Laws

1 ¬(p ∨ q) ≡ (¬p) ∧ (¬q)2 ¬(p ∧ q) ≡ (¬p) ∨ (¬q)

Properties

We have some properties for these operators and we can prove themusing truth tables.

1 Commutativity p ∧ q ≡ q ∧ p2 Associativity p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r3 Distribution p ∨ (q ∨ r) ≡ (p ∨ q) ∧ (p ∨ r)4 Idempotent p ∨ p ≡ p and p ∧ p ≡ p5 ¬(¬p) ≡ p

6 DeMorgan’s Laws1 ¬(p ∨ q) ≡ (¬p) ∧ (¬q)2 ¬(p ∧ q) ≡ (¬p) ∨ (¬q)

Properties

We have some properties for these operators and we can prove themusing truth tables.

1 Commutativity p ∧ q ≡ q ∧ p2 Associativity p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r3 Distribution p ∨ (q ∨ r) ≡ (p ∨ q) ∧ (p ∨ r)4 Idempotent p ∨ p ≡ p and p ∧ p ≡ p5 ¬(¬p) ≡ p6 DeMorgan’s Laws

1 ¬(p ∨ q) ≡ (¬p) ∧ (¬q)2 ¬(p ∧ q) ≡ (¬p) ∨ (¬q)

DeMorgan’s Laws

Example

Prove ¬(p ∨ q) ≡ (¬p) ∧ (¬q) using a truth table.

p q ¬p ¬q p ∨ q ¬(p ∨ q) (¬p) ∧ (¬q)T T F F T F FT F F T T F FF T T F T F FF F T T F T T

DeMorgan’s Laws

Example

Prove ¬(p ∨ q) ≡ (¬p) ∧ (¬q) using a truth table.

p q ¬p ¬q p ∨ q ¬(p ∨ q) (¬p) ∧ (¬q)T T F F T F FT F F T T F FF T T F T F FF F T T F T T

Other Types of Statements

Suppose p is the statement that the Red Sox score 5 runs and that q isthe statement that the Red Sox win.

1 If-Then p⇒ q

If the Sox score 5 runs then the Sox win.2 Converse q⇒ p

If the Sox win then they will score 5 runs.3 Inverse ¬p⇒ ¬q

If the Sox don’t score 5 runs then they don’t win.4 Contrapositive ¬q⇒ ¬p

If the Sox don’t win they did not score 5 runs.

Other Types of Statements

Suppose p is the statement that the Red Sox score 5 runs and that q isthe statement that the Red Sox win.

1 If-Then p⇒ qIf the Sox score 5 runs then the Sox win.

2 Converse q⇒ p

If the Sox win then they will score 5 runs.3 Inverse ¬p⇒ ¬q

If the Sox don’t score 5 runs then they don’t win.4 Contrapositive ¬q⇒ ¬p

If the Sox don’t win they did not score 5 runs.

Other Types of Statements

Suppose p is the statement that the Red Sox score 5 runs and that q isthe statement that the Red Sox win.

1 If-Then p⇒ qIf the Sox score 5 runs then the Sox win.

2 Converse q⇒ pIf the Sox win then they will score 5 runs.

3 Inverse ¬p⇒ ¬q

If the Sox don’t score 5 runs then they don’t win.4 Contrapositive ¬q⇒ ¬p

If the Sox don’t win they did not score 5 runs.

Other Types of Statements

Suppose p is the statement that the Red Sox score 5 runs and that q isthe statement that the Red Sox win.

1 If-Then p⇒ qIf the Sox score 5 runs then the Sox win.

2 Converse q⇒ pIf the Sox win then they will score 5 runs.

3 Inverse ¬p⇒ ¬qIf the Sox don’t score 5 runs then they don’t win.

4 Contrapositive ¬q⇒ ¬p

If the Sox don’t win they did not score 5 runs.

Other Types of Statements

Suppose p is the statement that the Red Sox score 5 runs and that q isthe statement that the Red Sox win.

1 If-Then p⇒ qIf the Sox score 5 runs then the Sox win.

2 Converse q⇒ pIf the Sox win then they will score 5 runs.

3 Inverse ¬p⇒ ¬qIf the Sox don’t score 5 runs then they don’t win.

4 Contrapositive ¬q⇒ ¬pIf the Sox don’t win they did not score 5 runs.

Biconditional Propositions

DefinitionLet p and q be propositions. The biconditional statement p↔ q is theproposition “p if and only if q”. The statement is true when p and qhave the same values and false otherwise.

p q p↔ qT T TT F FF T FF F T

Biconditional Propositions

DefinitionLet p and q be propositions. The biconditional statement p↔ q is theproposition “p if and only if q”. The statement is true when p and qhave the same values and false otherwise.

p q p↔ qT T TT F FF T FF F T

Operator Precedence

Operator Precedence

1 ¬2 ∧3 ∨ and ⊕ from left to right4 →

5 ↔

Operator Precedence

Operator Precedence

1 ¬2 ∧3 ∨ and ⊕ from left to right4 →5 ↔

IFF Example

ExampleDetermine the truth of the following statement if p is true and q isfalse:

(p→ q)↔ (¬q→ ¬p)

F ↔ F is true

Note: When a statement is true for all possible values of itspropositional variables, we have a tautology. A statement that isalways false is called a contradiction.

IFF Example

ExampleDetermine the truth of the following statement if p is true and q isfalse:

(p→ q)↔ (¬q→ ¬p)F ↔ F is true

Note: When a statement is true for all possible values of itspropositional variables, we have a tautology. A statement that isalways false is called a contradiction.

IFF Example

ExampleDetermine the truth of the following statement if p is true and q isfalse:

(p→ q)↔ (¬q→ ¬p)F ↔ F is true

Note: When a statement is true for all possible values of itspropositional variables, we have a tautology. A statement that isalways false is called a contradiction.

Converse

DefinitionThe converse of p→ q is q→ p.

The converse is used in proof writing in that we can prove p→ q istrue by proving q→ p is false.

ExampleState the converse of the following statements:

1 The Patriots win whenever it is snowing.We can rewrite this as the conditional proposition ‘If it issnowing, then the Patriots win’.The converse would therefore be ‘If the Patriots win then it issnowing’.

2 If it is raining then I will get wet.If I get wet then it is raining.

Converse

DefinitionThe converse of p→ q is q→ p.

The converse is used in proof writing in that we can prove p→ q istrue by proving q→ p is false.

ExampleState the converse of the following statements:

1 The Patriots win whenever it is snowing.We can rewrite this as the conditional proposition ‘If it issnowing, then the Patriots win’.The converse would therefore be ‘If the Patriots win then it issnowing’.

2 If it is raining then I will get wet.If I get wet then it is raining.

Converse

DefinitionThe converse of p→ q is q→ p.

The converse is used in proof writing in that we can prove p→ q istrue by proving q→ p is false.

ExampleState the converse of the following statements:

1 The Patriots win whenever it is snowing.

We can rewrite this as the conditional proposition ‘If it issnowing, then the Patriots win’.The converse would therefore be ‘If the Patriots win then it issnowing’.

2 If it is raining then I will get wet.If I get wet then it is raining.

Converse

DefinitionThe converse of p→ q is q→ p.

The converse is used in proof writing in that we can prove p→ q istrue by proving q→ p is false.

ExampleState the converse of the following statements:

1 The Patriots win whenever it is snowing.We can rewrite this as the conditional proposition ‘If it issnowing, then the Patriots win’.

The converse would therefore be ‘If the Patriots win then it issnowing’.

2 If it is raining then I will get wet.If I get wet then it is raining.

Converse

DefinitionThe converse of p→ q is q→ p.

The converse is used in proof writing in that we can prove p→ q istrue by proving q→ p is false.

ExampleState the converse of the following statements:

1 The Patriots win whenever it is snowing.We can rewrite this as the conditional proposition ‘If it issnowing, then the Patriots win’.The converse would therefore be ‘If the Patriots win then it issnowing’.

2 If it is raining then I will get wet.If I get wet then it is raining.

Converse

DefinitionThe converse of p→ q is q→ p.

The converse is used in proof writing in that we can prove p→ q istrue by proving q→ p is false.

ExampleState the converse of the following statements:

1 The Patriots win whenever it is snowing.We can rewrite this as the conditional proposition ‘If it issnowing, then the Patriots win’.The converse would therefore be ‘If the Patriots win then it issnowing’.

2 If it is raining then I will get wet.

If I get wet then it is raining.

Converse

DefinitionThe converse of p→ q is q→ p.

The converse is used in proof writing in that we can prove p→ q istrue by proving q→ p is false.

ExampleState the converse of the following statements:

1 The Patriots win whenever it is snowing.We can rewrite this as the conditional proposition ‘If it issnowing, then the Patriots win’.The converse would therefore be ‘If the Patriots win then it issnowing’.

2 If it is raining then I will get wet.If I get wet then it is raining.

Contrapositive

DefinitionThe contrapositive of p→ q is ¬q→ ¬p.

The contrapositive is logically equivalent to the original statement, sowe can prove a statement by proving it’s contrapositive.

ExampleState the contrapositive of the following statements:

1 The Patriots win whenever it is snowing.We can rewrite this as the conditional proposition ‘If it issnowing, then the Patriots win’.If the Patriots do not win then it is not snowing.

2 If it is raining then I will get wet.If I do not get wet then it is not raining.

Contrapositive

DefinitionThe contrapositive of p→ q is ¬q→ ¬p.

The contrapositive is logically equivalent to the original statement, sowe can prove a statement by proving it’s contrapositive.

ExampleState the contrapositive of the following statements:

1 The Patriots win whenever it is snowing.We can rewrite this as the conditional proposition ‘If it issnowing, then the Patriots win’.If the Patriots do not win then it is not snowing.

2 If it is raining then I will get wet.If I do not get wet then it is not raining.

Contrapositive

DefinitionThe contrapositive of p→ q is ¬q→ ¬p.

The contrapositive is logically equivalent to the original statement, sowe can prove a statement by proving it’s contrapositive.

ExampleState the contrapositive of the following statements:

1 The Patriots win whenever it is snowing.

We can rewrite this as the conditional proposition ‘If it issnowing, then the Patriots win’.If the Patriots do not win then it is not snowing.

2 If it is raining then I will get wet.If I do not get wet then it is not raining.

Contrapositive

DefinitionThe contrapositive of p→ q is ¬q→ ¬p.

The contrapositive is logically equivalent to the original statement, sowe can prove a statement by proving it’s contrapositive.

ExampleState the contrapositive of the following statements:

1 The Patriots win whenever it is snowing.We can rewrite this as the conditional proposition ‘If it issnowing, then the Patriots win’.

If the Patriots do not win then it is not snowing.2 If it is raining then I will get wet.

If I do not get wet then it is not raining.

Contrapositive

DefinitionThe contrapositive of p→ q is ¬q→ ¬p.

The contrapositive is logically equivalent to the original statement, sowe can prove a statement by proving it’s contrapositive.

ExampleState the contrapositive of the following statements:

1 The Patriots win whenever it is snowing.We can rewrite this as the conditional proposition ‘If it issnowing, then the Patriots win’.If the Patriots do not win then it is not snowing.

2 If it is raining then I will get wet.If I do not get wet then it is not raining.

Contrapositive

DefinitionThe contrapositive of p→ q is ¬q→ ¬p.

The contrapositive is logically equivalent to the original statement, sowe can prove a statement by proving it’s contrapositive.

ExampleState the contrapositive of the following statements:

1 The Patriots win whenever it is snowing.We can rewrite this as the conditional proposition ‘If it issnowing, then the Patriots win’.If the Patriots do not win then it is not snowing.

2 If it is raining then I will get wet.

If I do not get wet then it is not raining.

Contrapositive

DefinitionThe contrapositive of p→ q is ¬q→ ¬p.

The contrapositive is logically equivalent to the original statement, sowe can prove a statement by proving it’s contrapositive.

ExampleState the contrapositive of the following statements:

1 The Patriots win whenever it is snowing.We can rewrite this as the conditional proposition ‘If it issnowing, then the Patriots win’.If the Patriots do not win then it is not snowing.

2 If it is raining then I will get wet.If I do not get wet then it is not raining.

Logical Equivalence

When we have two statements that have the same truth valuesregardless of the truthfulness or falsehood of the propositions, wehave logical equivalence. Basically, we are saying that the statementsmay be different but they behave the same way.

Example

Determine the truth of the statements ¬(p ∨ q) and (¬p ∧ ¬q).

p q ¬(p ∨ q) (¬p ∧ ¬q)T TT FF TF F

p q ¬(p ∨ q) (¬p ∧ ¬q)T T FT F FF T FF F T

Logical Equivalence

When we have two statements that have the same truth valuesregardless of the truthfulness or falsehood of the propositions, wehave logical equivalence. Basically, we are saying that the statementsmay be different but they behave the same way.

Example

Determine the truth of the statements ¬(p ∨ q) and (¬p ∧ ¬q).

p q ¬(p ∨ q) (¬p ∧ ¬q)T TT FF TF F

p q ¬(p ∨ q) (¬p ∧ ¬q)T T FT F FF T FF F T

Logical Equivalence

When we have two statements that have the same truth valuesregardless of the truthfulness or falsehood of the propositions, wehave logical equivalence. Basically, we are saying that the statementsmay be different but they behave the same way.

Example

Determine the truth of the statements ¬(p ∨ q) and (¬p ∧ ¬q).

p q ¬(p ∨ q) (¬p ∧ ¬q)T TT FF TF F

p q ¬(p ∨ q) (¬p ∧ ¬q)T T FT F FF T FF F T

Logical Equivalence

When we have two statements that have the same truth valuesregardless of the truthfulness or falsehood of the propositions, wehave logical equivalence. Basically, we are saying that the statementsmay be different but they behave the same way.

Example

Determine the truth of the statements ¬(p ∨ q) and (¬p ∧ ¬q).

p q ¬(p ∨ q) (¬p ∧ ¬q)T TT FF TF F

p q ¬(p ∨ q) (¬p ∧ ¬q)T T FT F FF T FF F T

Final Table

p q ¬(p ∨ q) (¬p ∧ ¬q)T T F FT F F FF T F FF F T T

Notation: If P and Q are logically equivalent, we write P ≡ Q.

In our last example, we would say ¬(p ∨ q) ≡ (¬p ∧ ¬q).

Final Table

p q ¬(p ∨ q) (¬p ∧ ¬q)T T F FT F F FF T F FF F T T

Notation: If P and Q are logically equivalent, we write P ≡ Q.

In our last example, we would say ¬(p ∨ q) ≡ (¬p ∧ ¬q).

Final Table

p q ¬(p ∨ q) (¬p ∧ ¬q)T T F FT F F FF T F FF F T T

Notation: If P and Q are logically equivalent, we write P ≡ Q.

In our last example, we would say ¬(p ∨ q) ≡ (¬p ∧ ¬q).

Truth Table

Example

Construct a truth table for (p ∨ ¬q)⇒ (p ∧ q).

p q ¬q p ∨ ¬q p ∧ q (p ∨ ¬q)⇒ (p ∧ q)T T F T T TT F T T F FF T F F F TF F T T F F

Truth Table

Example

Construct a truth table for (p ∨ ¬q)⇒ (p ∧ q).

p q ¬q p ∨ ¬q p ∧ q (p ∨ ¬q)⇒ (p ∧ q)T T F T T TT F T T F FF T F F F TF F T T F F

If-Then and Contrapositive

The relationship between the if-then statement and the contrapositivestatement is important in that they are logically equivalent.

p q ¬p ¬q p⇒ q ¬q⇒ ¬pT T F F T TT F F T F FF T T F T TF F T T T T

This is an example of a contingency since we get some of theoutcomes to be true and some to be false.

If-Then and Contrapositive

The relationship between the if-then statement and the contrapositivestatement is important in that they are logically equivalent.

p q ¬p ¬q p⇒ q ¬q⇒ ¬pT T F F T TT F F T F FF T T F T TF F T T T T

This is an example of a contingency since we get some of theoutcomes to be true and some to be false.

If-Then and Contrapositive

The relationship between the if-then statement and the contrapositivestatement is important in that they are logically equivalent.

p q ¬p ¬q p⇒ q ¬q⇒ ¬pT T F F T TT F F T F FF T T F T TF F T T T T

This is an example of a contingency since we get some of theoutcomes to be true and some to be false.