Conditional Statements

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Presentation transcript:

Conditional Statements

What is a conditional statement? A conditional statement is any statement that can be expressed in the form “if p, then q”, where p is the hypothesis, and q is the conclusion Notation: p  q

If you are funny then you will make people laugh. For each example, rewrite the statement in “if…then...” and identify the hypothesis and conclusion. 1. You will make people laugh if you are funny. If you are funny then you will make people laugh. hypothesis conclusion

For each example, rewrite the statement in “if…then For each example, rewrite the statement in “if…then...” and identify the hypothesis and conclusion. 2. A line which bisects a segment contains the midpoint of the segment. If a line bisects a segment then it contains the midpoint of the segment. hypothesis conclusion

If a person is left-handed then he is male. For each example, rewrite the statement in “if…then...” and identify the hypothesis and conclusion. 3. All left-handed people are male. If a person is left-handed then he is male. hypothesis conclusion

If a number ends in 5 then it is odd. For each example, rewrite the statement in “if…then...” and identify the hypothesis and conclusion. 4. A number is odd whenever it ends in 5. conclusion If a number ends in 5 then it is odd. hypothesis

When is a conditional statement false? Ex: Bobby makes this statement to his girlfriend: “If I earn $20 cutting grass, then I will take you to the movies.” Earns $20 (E) Takes her to movie (M) E  M T F Ok - T NOT Ok - FALSE Technically statement does not apply

When is a conditional statement false? Ex: Bobby makes this statement to his girlfriend: “If I earn $20 cutting grass, then I will take you to the movies.” Earns $20 (E) Takes her to movie (M) E  M T F Ok - T NOT Ok - FALSE Ok - T DEFINED VALUES Ok - T

When is a conditional statement false? A conditional statement is only false if the hypothesis is true and the conclusion is false. It is true in all other cases. Earns $20 (E) Takes her to movie (M) E  M T F

When is a conditional statement false? A conditional statement is only false if the hypothesis is true and the conclusion is false. It is true in all other cases. ~(p  q)  p  ~q

What is a biconditional statement? A biconditional statement is composed of two propositions joined by the phrase “if and only if”. Example: This is period 7 if and only if this is Geometry. Notation: p  q  (p  q)  (q  p)

When is a biconditional statement true When is a biconditional statement true? When is a biconditional statement false? p  q  (p  q)  (q  p) Expression is false when p is true and q is false; true in all other cases Expression is false when q is true and p is false; true in all other cases p q p  q q  p T F

When is a biconditional statement true When is a biconditional statement true? When is a biconditional statement false? A biconditional statement is true when both propositions have the same truth value (both true or both false). A biconditional statement is false when the propositions have different truth values (one is the true, the other false) p q p  q q  p T F

What is the negation of a biconditional? Remember: p  q  (p  q)  (q  p) ~(p  q)  ~(p  q)  ~(q  p) Negation: ~(p  q)  (p  ~q)  (q  ~p)