If Then If then If then Conditional Statements

If…. Then……

If …then…

If…. then…. .

Conditional Statements • Written in the form if “a” then “b” means if “a” happens then “b” WILL happen • • In real life: If you clean your room then you can go out. Ex. If a polygon is a pentagon then it has 5 sides • There are two parts: hypothesis Conclusion • • - if “a” then “b” Need to evaluate the conditional statements as true or false. If “a” happens then “b” must always follow. To find counterexamples try to find where the “a” part happens but the “b” does not. If you can find even one counterexample then the statement is not true!!!

More Examples • Ex. If the sum of two numbers is even the two numbers are both even To find this counterexample, look for 2 numbers whose sum is even but the two numbers are not even This is false since 4 = 3 + 1 • Ex. If you live in Edmonton Alberta, then you live in the capital of Alberta. True: since there is no way for the first part to be true and the second part to be false.

Converse • A conditional statement with the hypothesis and conclusion interchanged. If “b” then “a” • Ex. 1 st conditional statement: If a polygon is a pentagon, then , the polygon has 5 sides Converse: If the polygon has 5 sides, then a polygon is a pentagon. NOTE: feel free to fix grammer issues that might arise ( like “the”, “a”, verb tenses, contractions, ect.

How is this a converse?

Bi. Conditional • If a conditional statement and its converse are both true, they can be combined into an if and only if statement called a biconditional • “a” if and only if “b” if and only if “a” (the order in this doesn’t matter) • Ex. A polygon is a pentagon if and only if it has 5 sides.

Contrapositive • The negation of both parts of the converse • If not “b” then not “a” • Ex. If a polygon does not have 5 sides then the polygon is not a pentagon. • Ex. If a triangle has a 90°angle then it is a right triangle. (is this True of False? ) WHAT ARE THE: Converse, Biconditional, and contrapostitive? ? ? - Converse: if it is a right triangle then a triangle has a 90° angle. - Biconditional: A triangle is a right triangle if and only if it has a 90 ° angle. OR a triangle has a 90 ° angle if and only if it is a right triangle - Contrapositive: If it is not a right triangle then a triangle does not have 90 ° angle.

More Examples • Ex. If angle A= 30° then angle A is acute (T or F) • Converse? If angle A is acute then angle A= 30 ° • Biconditional? N/A (don’t even write it out unless both the original and the converse are true) • Contrapositive? • If angle A is not acute then Angle A ≠ 30 ° (is this T or F)

• Ex. If x>0 then x²>0 (T or F) - Converse? If x²>0 then x>0 (T or F) an example would be 9>0 then (-3)²>0 but (-3 is NOT>0) - Biconditional? N/A (because the converse and the original are different and not BOTH true) - Contrapositive? If x² is NOT> 0 then x is NOT> 0 Think of it this way: if x² is ≤ 0 then x ≤ 0. This is because since x²< 0 is impossible then x=0 for x²≤ 0 AND x ≤ 0 is true if x=0 since 0 ≤ 0

Examples • If an integer is divisible by 2 then it is not odd • Converse? If an integer is not odd then it is divisible by 2 ( T or F) • Biconditional? An integer is divisible by 2 if and only if it is not odd. (T or F) • Contrapositive? If an integer is not odd then it is not divisible by 2 (since the above is a double negative: If an integer is odd then it is not divisible by 2)

The Gossiping Defenders • Pg. 62