Logic and Reasoning. Identify the hypothesis and conclusion of each conditional. Example 1: Identifying the Parts of a Conditional Statement A.If today.

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Logic and Reasoning

Identify the hypothesis and conclusion of each conditional. Example 1: Identifying the Parts of a Conditional Statement A.If today is Thanksgiving Day, then today is Thursday. B. If a number is an integer, then it is a rational number. Hypothesis: Today is Thanksgiving Day. Conclusion: Today is Thursday. Hypothesis: A number is an integer. Conclusion: The number is a rational number.

DefinitionSymbols The converse is the statement formed by exchanging the hypothesis and conclusion. “If q, then p.” q  pq  p DefinitionSymbols The inverse is the statement formed by negating the hypothesis and conclusion. “If not p, then not q.” ~p  ~q

DefinitionSymbols The contrapositive is the statement formed by exchanging the hypothesis and conclusion and then negating them. “If not q, then not p.” (Taking both the converse and the inverse) ~q  ~p

* Conditional p  q * If I studied, then I did well in Geometry. * Converse q  p * If I did well in Geometry, then I studied. * Inverse ~p  ~q * If I did not study, then I did not do well in Geometry. * Contrapositive ~ q  ~p * If I did not do well in Geometry, then I did not study.

* Write 8 sets of conditional statements like the last example. Each set should have 4 sentences: conditional, converse, inverse, and contrapositive. Be sure to label each sentence with what type of logical statement it is. * This must be finished and turned in by the end of class today.

* Not all logic statements are true. Oftentimes, if the conditional statement is true, then the converse is not true. * Example) * Conditional: If you are a human, then you have hair. * Converse: If you have hair, then you are human.

* A counterexample shows that the conclusion of a logic statement is false. * If you have hair, then you are human. * Counterexample: bears have hair

* If a conditional statement and its converse are both true, we call that statement a bi-conditional statement. * Conditional: If there is thunder, then there is lightning. * Converse: If there is lightning, then there is thunder. * Bi-conditional: There is thunder if and only if there is lightning.

A bi-conditional statement is written in the form “p if and only if q.” This means “if p, then q” and “if q, then p.” p q means p q and q p Symbols Bi-conditional Conditional Converse

* Conditional: If a figure has three sides, then it is a triangle. * Converse: If a figure is a triangle, then it has three sides. * Bi-conditional: