And represent them using Venn Diagrams.

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Presentation transcript:

And represent them using Venn Diagrams. Geometry Lesson 2 – 2 Logic Objective: Determine the truth values of negations, conjunctions, and disjunctions And represent them using Venn Diagrams. Find counterexamples.

Logic p: A rectangle is a quadrilateral Truth value of p: T Statement – a sentence that is either true or false Often represented by letters such as p or q. Truth value – the value of a statement either T or F p: A rectangle is a quadrilateral Truth value of p: T

Negation ~p: A rectangle is not a quadrilateral. Negation – negation of a statement has the opposite meaning as well as opposite truth value. ~p read ‘not p’ ~p: A rectangle is not a quadrilateral.

Statements Compound statement Conjunction Disjunction Two or more statements joined by the word and or or Conjunction Compound statement using and Symbol Disjunction Compound statement using or

Conjunction p: True q: True p: A rectangle is a quadrilateral q: A rectangle is convex. What is the truth value of p q? p: True q: True p and q : True

F p: T r: F T True and True = True Q: T r: F ~r: T Use the following statements to write a conjunction. Then find its truth value. Explain your reasoning. p: The figure is a triangle q: The figure has two congruent sides r: The figure has three acute angles. p and r F p: T r: F Since both statements are not T The whole statement is False T True and True = True Q: T r: F ~r: T

Cont… T p: The figure is a triangle q: The figure has two congruent sides r: The figure has three acute angles. not p and not r T True and true F False and true

Disjunction p q p: Malik studies geometry q: Malik studies chemistry p or q: Malik studies geometry or chemistry. p q

F q or r q: F r: T False or True T p: January is a fall month Use the following statements to write a conjunction. Then find its truth value. Explain your reasoning. p: January is a fall month q: January has only 30 days r: January 1 is the first day of a new year. q or r q: F r: T False or True T F False or False P: F Q: F

T P: false, ~p: true R: true r or p T F T True or true p: January is a fall month q: January has only 30 days r: January 1 is the first day of a new year. T r or p P: false, ~p: true R: true True or true T r: True, p: false F q: false, ~r: false T p: false, ~q: true

Truth Table A way of organizing truth values T F F T

Truth Table: Conjunction F F F T F F F F Conjunction is only T when both are True

Truth Table: Disjunction F T F T T F F F Disjunction is only False when both are false.

Construct a Truth Table p q ~p T T F T T F F F F T T T F F T T

Truth Table p q ~p ~q T T F F F T F T T T F T T F F F F T T F

Construct a truth table p q r ~q T T T F T T T T F F F F T F F T T T T F T T T T F T T F T F F T F T T F F F F F T F

Venn Diagrams The Venn diagram shows the number of graduates last year who did or did not attend their junior or senior prom. How many graduates attended their senior but not their junior Prom? How many graduates attended their junior and senior proms? 25 123

37 How many graduates did not attend either of their proms? How many students graduated last year? Explain your reasoning. 37 270 students 85 that went as Juniors, 25 that went as Seniors, 123 that went to both 37 that did not attend either..

Homework Pg. 101 1 – 10 all, 12 – 30 E, 34, 36, 38