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.

Definition Symbols

The converse is the statement formed by exchanging the hypothesis and conclusion. “If q, then p.”

q p

Definition Symbols

The inverse is the statement formed by negating the hypothesis and conclusion. “If not p, then not q.”

~p ~q

Definition Symbols

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 pSymbols

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: