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Section 1.1

Logic

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Proposition

• Statement that is either true or false– can’t be both– in English, must contain a form of “to be”

• Examples:– Cate Sheller is President of the United States– CS1 is a prerequisite for this class– I am breathing

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Many statements are not propositions ...

• Give me liberty or give me death

• ax2 + bx + c = 0

• See Spot run

• Who am I and why am I here?

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Representing propositions

• Can use letter to represent proposition; think of letter as logical variable

• Typically use p to represent first proposition, q for second, r for third, etc.

• Truth value of a proposition is T (true) or F(false)

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Negation

• Logical opposite of a proposition

• If p is a proposition, not p is its negation

• Not p is usually denoted:

p

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Truth table

• Graphical display of relationships between truth values of propositions

• Shows all possible values of propositions, or combinations of propositions

p p

T FF T

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Logical Operators

• Negation is an example of a logical operation; the negation operator is unary, meaning it operates on one logical variable (like unary arithmetic negation)

• Connectives are operators that operate on two (or more) propositions

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Conjunction

• Conjunction of 2 propositions is true if and only if both propositions are true

• Denoted with the symbol • If p and q are propositions, p q means p

AND q

• Remember - looks like A for And

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Examples Let p = 2 + 2 = 4

q = “It is raining”r = “ I am in class now”

What is the value of:p q p rr q p r p r (p r)

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Truth table for p q

p q p q

T T TT F FF T FF F F

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Disjunction

• Disjunction of two propositions is false only if both propositions are false

• Denoted with this symbol: • If p and q are propositions, p q means p

OR q

• Mnemonic: looks like OAR in the water (sort of)

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Examples Let p = 2 + 2 = 4

q = “It is raining”r = “ I am in class now”

What is the value of:p q p rr q p r p r (p r)

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Truth table for disjunction

p q p q

T T TT F TF T TF F F

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Inclusive vs. exclusive OR

• Disjunction means or in the inclusive sense; includes the possibility that both propositions are true, and can be true at the same time

• For example, you may take this class if you have taken Calculus I or you have the instructor’s permission - in other words, you can take it if you have either, or both

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Exclusive OR

• The exclusive or of two propositions is true when exactly one of the propositions is true, false otherwise

• Exclusive or is denoted with this symbol: • For p and q, p q means p XOR q

• Mnemonic: looks like sideways X inside an O

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English examples

• I am either in class or in my office

• The meal comes with soup or salad

• You can have your cake or you can eat it

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Examples Let p = 2 + 2 = 4

q = “It is raining”r = “ I am in class now”

What is the value of:p q p rr q p r p r (p r)

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Truth table for

p q p q

T T FT F TF T TF F F

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Implication

• The implication of two propositions depends on the ordering of the propositions

• The first proposition is calls the premise (or hypothesis or antecedent) and the second is the conclusion (or consequence)

• An implication is false when the premise is true but the conclusion is false, and true in all other cases

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Implication• Implication is denoted with the symbol • For p and q, p q can be read as:

– if p then q– p implies q– q if p– p only if q– q whenever p– q is necessary for p– p is sufficient for q– if p, q

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Implication

• Note that a false premise always leads to a true implication, regardless of the truth value of the conclusion

• Implication does not necessarily mean a cause and effect relationship between the premise and the conclusion

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Implications in English

• If Cate lives in Iowa, then Discrete Math is a 3-credit class

• Since p (I live in Iowa) and q (this is a 3-credit class) are both true, p q is true even though p and q are unrelated statements

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Implications in English

• If the sky is brown, then 2+2=5

• Since p (sky is brown) and q (2+2=5) are both false, the implication p q is true

• Remember, you can conclude anything from a false premise

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If/then vs. implication

• In programming, the if/then logic structure is not the same as implication, though the two are related

• In a program, if the premise (if expression) is true, the statements following the premise will executed, otherwise not

• There is no “conclusion,” so it’s not an implication

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Examples Let p = 2 + 2 = 4

q = “It is raining”r = “ I am in class now”

What is the value of:p q p rr q p r p r (p r)

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Truth table for

p q p q

T T TT F FF T TF F T

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Converse & contrapositive

• For the implication p q, the converse is q p

• For the implication p q, the contrapositive is q p

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Biconditional

• A biconditional is a proposition that is true when p and q have the same truth values (both true or both false)

• For p and q, the biconditional is denoted as p q, which can be read as:– p if and only if q– p is necessary and sufficient for q– if p then q, and conversely

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Examples Let p = 2 + 2 = 4

q = “It is raining”r = “ I am in class now”

What is the value of:p q p rr q p r p r (p r)

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Truth table for

p q p q

T T TT F FF T FF F T

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Compound propositions

• Can build compound propositions by combining simple propositions using negation and connectives

• Use parentheses to specify order or operations

• Negation takes precedence over connectives

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Examples Let p = 2 + 2 = 4

q = “It is raining”r = “ I am in class now”

What is the value of:(p q) ( p r)(r q) (p r) (p r ) (p r)

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Logic & Bit Operations

• A bit string is a sequence of 1s and 0s - the number of bits in the string is the length of the string

• Bit operations correspond to logical operations with 1 representing T and 0 representing F

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Bit operation examples

Let s1 = 10011100 s2 = 11000110

s1 OR s2 = 11011110s1 AND s2 = 10000100s1 XOR s2 = 01011010

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Section 1.1

Logic

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