12.2: And and Or in Logic04/10/23

12.2: And, Or and Not in Logic

Expectations:L3.2.1:Know and use the terms of basic logic (e.g.,

proposition, negation, truth and falsity, implication, if and only if, contrapositive, and converse).

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.

L3.2.3:Use the quantifiers “there exists” and “all” in mathematical and everyday settings and know how to logically negate statements involving them.

12.2: And and Or in Logic04/10/23

Statements

Statement in Logic: Sentence that is either true or false.

Compound Statement: 2 or more statements connected together usually by “and” or “or.”

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Conjunction:

Compound statement using “_____” as the connector.

“p ____ q”

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Ex: A square is a rhombus and it is a rectangle.

p: a square is a rhombusq: a square is a rectangle

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A conjunction is true iff p and q are ___________.

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Truth table for conjunctions.

p q

true true

true false

false true

false false

p and q

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Is the following conjunction true or false?

“Parallel lines do not intersect and perpendicular lines do intersect.”

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Disjunction:

Compound statement using “____” as the connector.

“p ____ q”

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Exclusive or: one or the other but not both.

“Betty is right handed or she is left handed.”

Inclusive or: one or the other or both.

“It may be cloudy or cool tomorrow.”

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Disjunctions use the ________ or.

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Truth table for disjunctions.

p q

true true

true false

false true

false false

p or q

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Is the following disjunction true or false?“A rectangle has at least 2 right angles or a square is a closed 3 sided figure.”

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Quantifiers

Three common geometric quantifiers are “there exists,” “all” and “every.”

There exists just means ______________ of something exists.

All and every basically mean the same thing: each and every thing must satisfy the condition.

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Given a plane and a line in the plane, there exists a point in the plane that is not on the line.

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All lines contain at least 2 points.

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Negating Quantifiers

What is the negation of this statement?

There exists an angle whose measure is less than 90 degrees.

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What is the negation of this statement?

“All angles have 2 sides.”

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What can you say about the negation of a “there exists” statement?

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What can you say about the negation of an “all” or “every” statement.

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Write the negations of the statements below.

“There exists a smallest real number.”

“There exists a 2 sided polygon.”

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Write the negations of the statements below.

“All triangles are polygons.”

“Every mammal is a dog.”

Identify the negation of the statement below.

No quadrilaterals have congruent angles and congruent sides.

A. All quadrilaterals have congruent angles and congruent sides.

B. Squares have congruent angles and congruent sides.

C. Rectangles have congruent angles but not congruent sides.

D. There exists a quadrilateral with congruent sides and congruent angles.

E. There does not exist a quadrilateral with congruent sides and angles.

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Construct a truth table for (p and q) or r

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Complete a truth table for ~(p and q) and a separate truth table for ~p or ~q

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~(p and q) is truth functionally equivalent to ~p or ~q.

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Assignment:

pages 779-781,

# 11 - 35 (all).