04/19/23 2.2: An Introduction to Logic

2.2: An Introduction to LogicExpectations:L3.2.1: Know and use the terms of basic logic.L3.2.2: Use the connectives “not,” “and,” “or,” and

“if..., then,” in mathematical and everyday settings. Know the truth table of each connective and how to logically negate statements involving these connectives.

L3.2.4: Write the converse, inverse, and contrapositive of an “If..., then...” statement. Use the fact, in mathematical and everyday settings, that the contrapositive is logically equivalent to the original while the inverse and converse are not.

04/19/23 2.2: An Introduction to Logic

Conditional Statements

“If I get an A on my test, then I get to go to Disneyworld.”

“If the sky is blue, then it is not raining.”

“If x2 = 16, then x = 4.”

04/19/23 2.2: An Introduction to Logic

Conditional Statements

“If p, then q.”

p: _______________

q: _______________

“If” is NOT a part of the hypothesis and “then” is NOT part of the conclusion.

The hypothesis and conclusion are each complete sentences.

04/19/23 2.2: An Introduction to Logic

Conditional Statements

Identify the hypothesis and conclusion for the statement below.

If I get all of my work done, then I get to play.

Hypothesis:

Conclusion:

04/19/23 2.2: An Introduction to Logic

Conditional Statements

Logic Form

p q

“ p implies q”

04/19/23 2.2: An Introduction to Logic

Conditional Statements

Write, “If x = 3, then x + 4 = 7,” in logic form.

04/19/23 2.2: An Introduction to Logic

Other Forms of Conditionals

Instead of being written using the terms if and then, some conditionals are written using terms like, all, every and q if p.

All squares are rectangles.Every fraction is a real number.Angles are supplementary if they

form a linear pair.Vertical angles are congruent.

04/19/23 2.2: An Introduction to Logic

It can be very helpful to rewrite these other forms as if-then statements.

04/19/23 2.2: An Introduction to Logic

Write the following in if – then form.

a. All squares are rectangles.

b. Every cat is a mammal.

c. We will not have school if we get 12 inches of snow.

04/19/23 2.2: An Introduction to Logic

Conditional Statements

Venn (Euler) Diagrams for Conditionals

___

__

“If p, then q.”

p=>q

04/19/23 2.2: An Introduction to Logic

Draw a Venn Diagram for:

If a student is a freshman (9th grade), then they are in high school.

04/19/23 2.2: An Introduction to Logic

Write a Conditional Statement for:

A triangle is isosceles.

A triangle is equilateral.

04/19/23 2.2: An Introduction to Logic

Conditional Statement: p q. Conditional statements as promises:

If p happens, then “I promise” q will happen.

04/19/23 2.2: An Introduction to Logic

Ex: If Trevor eats a good dinner, then he gets dessert.

If Trevor eats a good dinner, then I promise he gets dessert.

04/19/23 2.2: An Introduction to Logic

Truth Value of Conditional Statements:

p q is false iff the promise is broken.

04/19/23 2.2: An Introduction to Logic

If I wash the car, then I get to go to the movies.

4 cases to consider: 1. I wash the car and I go to the movies.

04/19/23 2.2: An Introduction to Logic

2. I wash the car, but do not get to go to the movies.

3. I do not wash the car, but I go to the movies.

04/19/23 2.2: An Introduction to Logic

4. I do not wash the car and I do not get to go to the movies.

Bellwork 9/29/2010

S is between A and B such that AS is 4 more than SB tripled. If AB = 52, what are the AS and SB?

04/19/23 2.2: An Introduction to Logic

Truth Table for Conditionals.

p q

true true

true false

false true

false false

p q

04/19/23 2.2: An Introduction to Logic

Converses

The converse of p q is ________.

p q: If x + 4 = 10, then x = 6.

q p: If x = 6, then x + 4 = 10.

04/19/23 2.2: An Introduction to Logic

Write a conditional with the hypothesis “a triangle is a right triangle” and the conclusion “it has a 90° angle.”

04/19/23 2.2: An Introduction to Logic

Write the converse of your conditional. Is it true or false?

04/19/23 2.2: An Introduction to Logic

Determine the truth value of the conditional below. Then write the converse and

determine its truth value.

If M is the midpoint of AB, then AM = MB.

04/19/23 2.2: An Introduction to Logic

04/19/23 2.2: An Introduction to Logic

Truth Table for Converses.

p q

true true

true false

false true

false false

q p

ACT/PLAN Prep

When the point (4,-1) is reflected across the y-axis, what are the coordinates of its image?

A. (-4,-1)B. (-4,1)C. (-1,4)D. (4,-1)E. (4,1)

04/19/23 2.2: An Introduction to Logic

Assignment 2.2.1

pages 95 – 98, 9 – 12 (all), 14 – 28 (evens)

04/19/23 2.2: An Introduction to Logic

Functional Equivalence

Two statements are said to be “logically equivalent,” “truth functionally equivalent” or simply “functionally equivalent” if there truth tables are identical. The tables must be in the same order.

04/19/23 2.2: An Introduction to Logic

Are conditional statements and their converses functionally equivalent?

04/19/23 2.2: An Introduction to Logic

Negation:

The nullifying of a statement.

The negation of statement p is not p, written ______.

04/19/23 2.2: An Introduction to Logic

Examples:p: today is sunny

~p:

p: x = 4

~p:

p: Marcus is not late

~p:

04/19/23 2.2: An Introduction to Logic

A statement and its negation have __________ truth values.

04/19/23 2.2: An Introduction to Logic

Truth table for negations.

p

true

false

~p

04/19/23 2.2: An Introduction to Logic

Inverse of a Conditional Statement:

The inverse of p q is __________.

04/19/23 2.2: An Introduction to Logic

p q : “If a quadrilateral is a square, then it is a rhombus.”

Write the inverse and determine the truth value of both statements.

04/19/23 2.2: An Introduction to Logic

04/19/23 2.2: An Introduction to Logic

p q : “If a triangle is equilateral, then all of its angles are congruent”

Write the inverse and determine the truth value of both statements.

04/19/23 2.2: An Introduction to Logic

04/19/23 2.2: An Introduction to Logic

Truth Table for Inverses.

p q

true true

true false

false true

false false

~p ~q ~p ~q

04/19/23 2.2: An Introduction to Logic

Are conditionals and their inverses functionally equivalent?

04/19/23 2.2: An Introduction to Logic

Contrapositive of a Conditional Statement:

The contrapositive of p q is ___________.

04/19/23 2.2: An Introduction to Logic

p q : If a quadrilateral is a square, then it is a rectangle.

Write the contrapositive and determine the truth value of both

statements.

04/19/23 2.2: An Introduction to Logic

04/19/23 2.2: An Introduction to Logic

Truth Table for Contrapositives.

p q

true true

true false

false true

false false

~p ~q ~q ~p

04/19/23 2.2: An Introduction to Logic

Are conditionals and their contrapositives functionally equivalent?

04/19/23 2.2: An Introduction to Logic

Are converses functionally equivalent to inverses?

Consider the following statement: “If m is an odd number, then m is not divisible by 2.”

Which represents the contrapositive of the original statement?

A. If m is divisible by 2, then m is an odd number.

B. If m is divisible by 2, then m is not an odd number.

C. If m is an odd number, then m is not divisible by 2.

D. If m is not divisible by 2, then m is not an odd number.

04/19/23 2.2: An Introduction to Logic

04/19/23 2.2: An Introduction to Logic

Transitive Property for Conditionals

If p q and q r, then ___________.

04/19/23 2.2: An Introduction to Logic

Make a conclusion given the following statements:

If a triangle is equilateral, then it is isosceles.

If a triangle is isosceles, then it has at least 2 congruent angles.

04/19/23 2.2: An Introduction to Logic

Logic Chains

The transitive property can be extended past 2 statements to form a logic chain.

Ex: If a b, b c, c d, d e, e f, then _______.

04/19/23 2.2: An Introduction to Logic

Using the following statements, what may you conclude?

If a polygon is a square then it is a rectangle.

If a polygon is a rectangle, then it is a parallelogram.

If a polygon is a parallelogram, then it is a quadrilateral.

04/19/23 2.2: An Introduction to Logic

Using the following statements, what may you conclude?

If a quadrangle is a square then it is a rhombus.

If a quadrangle is a parallelogram, then it has exactly 2 pairs of parallel opposite sides.

If a quadrangle is a rhombus, then it is a parallelogram.

Identify the statement that must be logically equivalent to the given

statement.“If a bird is a cardinal, then it is red.”

A. If a bird is not a cardinal, then it is not red.

B. If a bird is red, then it is a cardinal.

C. If a bird is red, then it is not a cardinal.

D. If a bird is not red, then it is a cardinal.

E. If a bird is not red, then it is not a cardinal.

04/19/23 2.2: An Introduction to Logic

Assignment 2.2.2

pages 95 – 98, #9 – 12 (all), 30, 32 and 51 – 53 (all)

Pages 788 – 790, #10, 12, 14, 22 and 24