Notes Conditional Statement: is a logical statement with two parts, a hypothesis and a conclusion. Hypothesis: are the conditions that we’re considering Conclusion: is what follows as a result of the conditions in the hypothesis. If-then form: a style of stating a conditional statement where the hypothesis comes immediately after the word if and the conclusion comes immediately after the word then.
Notes Example: If-then form of a conditional statement: If it is raining outside, then the ground is wet. Hypothesis – it is raining outside Conclusion – the ground is wet.
Notes On your own: Identify the hypothesis and the conclusion, then write the following conditional statement in if-then form. A number divisible by 9 is divisible by 3 h: _____________________________ c: _____________________________ If ________________, then _______________ The 49ers will play in the Super Bowl XLII, if they win their next game. h: _____________________________ c: _____________________________ If _________________ then ________________
Notes For a conditional statement to be true, it must be proven true for all cases that satisfy the conditions of the hypothesis A single counterexample is enough to prove a conditional statement false On Your Own: Write a counterexample to show that the following statement is false. If x 2 = 16, then x = 4 Counterexample: _______________________ This proves the statement false.
Notes Related Conditionals: other statements formed by changing the original statement. Converse: of a statement is formed by switching the conclusion and the hypothesis. The converse of a statement is not always true! Example: Original: If it’s raining outside, then the ground is wet. Converse: If the ground is wet, then it is raining outside. Q: is the converse true or false? Answer: ____________
Notes On Your Own: Write the converse of the following statement Original: If a number is divisible by 9 then it is divisible by 3 Converse: _____________________________ Q: is the converse true or false? Answer: ___________ Original: If two segments are congruent, then they have the same length. Converse: _____________________________ Q: is the converse true or false? Answer: ___________
Notes Inverse: formed by negating the hypothesis and conclusion of the statement Example Statement: If it’s raining outside, then the ground is wet. Inverse: If it’s not raining outside, then the ground is not wet. On Your Own: write the inverse of the following statement Statement: If two segments are congruent, then they have the same length. Inverse: _______________________________
Notes Contrapositive: (Combination of converse and inverse) formed by switching the hypothesis and the conclusion and negating them. Example Statement: If it’s raining outside, then the ground is wet. Contrapositive: if the ground is not wet, then it is not raining outside On Your Own Statement: If two segments are congruent, then they have the same length. Contrapositive: _____________________________
Notes Logically Equivalent Statements: Statements that have the same truth value (i.e. when one is true, so is the other) A statement and its contrapositive are equivalent statements Original: If it’s raining outside, the ground is wet. Contrapositive: If the ground isn’t wet, does that mean it isn’t raining? Yes Lets think about this, are these two saying the same thing? The converse and inverse are also logically equivalent.
Conditional Statement Activity Come up with your own conditional statement in if-then form Write the converse, inverse, and contrapositive. Judge the validity of all four statements. Do the equivalent statements match up as they should and make sense? Write counterexamples for the statements you think are false. Be sure to: Label the four statements, Indicate whether each is true or false, and Show which statements are equivalent to each other.