If-then statements April 15, 2008 What is an if-then statement? • One of the postulates we looked at earlier stated: If B is between A and C, then AB + BC = AC. • This type of statement is known as an ifthen statement for obvious reasons. We also refer to them as conditional statements or just conditionals. All about conditionals • A conditional consists of two parts; a hypothesis and a conclusion. • The hypothesis is the part of the statement immediately after the “if,” which we will represent by the letter p. • The conclusion is the part of the statement after the “then,” which we will represent by the letter q. Writing if then statements using p and q • You can write the conditional statement, “if p then q” like this: p→q What is the converse? • The converse of a conditional is formed by interchanging the hypothesis and the conclusion. • So, instead of if p then q, the converse is if q then p. • We can write this as q→p If the conditional is true, is the converse true? • If you are in Ameena’s math class, then you attend CMC. • What is the converse? • To prove a conditional statement false all we need to do is find one counterexample. What if both the conditional and the converse are true? • If both a conditional and its converse are true (we never said it wasn't possible!) we can combine them into a single statement using the words "if and only if." What is the inverse? • If we negate the hypothesis and the conclusion we get the inverse. • Instead of if p, then q, we have if not p, then not q. • We can write this as: ~p → ~q What is the contrapositive? • Finally, if we interchange and negate the hypothesis and conclusion we get the contrapositive. • The contrapositive of "If p, then q" is "If not q, then not p.“ • We write this as: ~q→~p What is logical equivalence? • When two statements always have the same truth value we say that they are logically equivalent. • So a conditional and its contrapositive are logically equivalent as are the converse and the inverse. • . What kinds of logical equivalence are possible? • All four may be true or all four may be false. • Two of the statements might be true and two might be false but these are the only possibilites. • It is not possible for three of the statements to be true and the last one false or for three of the statements to be false and the last one true. To review • • • • Conditional: p→q converse: q→p inverse: ~p→~q contrapositive: ~q→~p

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