Lesson PROOF THAT THE INVERSE AND CONVERSE OF THE IMPLIES STATEMENT ARE EQUIVALENT
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This proof uses the law of contra-position to prove that the inverse of the implies statement is equivalent to the converse of the implies statement. It follows that with the use of the law of equivalence to prove that the inverse of the implies statement is equivalent to the converse of the implies statement. The law of contra-position states: (p -> q) is equivalent to (~q -> ~p) The law of contra-position has already been proven. Here's a link to that proof: <a href = "http://www.algebra.com/algebra/homework/Conjunction/THEO-2011-08-22-01.lesson" target = "_blank">PROOF OF THE LAW OF CONTRA-POSITION</a> To prove that the inverse of the implies statement is equivalent to the converse of the implies statement, all we have to do is show that the converse of the implies statement is the contra-positive of the inverse of the implies statement. We'll start with the implies statement itself. Symbolically that looks like: p -> q The inverse of the implies statement looks like: ~p -> ~q The converse of the implies statement looks like: q -> p If we let: a = ~p b = ~q Then: ~a = p ~b = q And we get: The inverse of the implies statement looks like: a -> b The converse of the implies statement looks like: ~b -> ~a This shows that the converse of the implies statement is the contra-positive of the inverse of the implies statement. Since this is true, then the law of contra-position applies and the two statements are equivalent. We can also prove they are equivalent by using the law of equivalency. The proof table for the implies statement and the inverse of the implies statement and the converse of the implies statement is shown below: <pre> 1 2 3 4 5 6 7 8 p q ~p ~q (p -> q) (~p -> ~q) (q -> p) (~p -> ~q) <-> (q -> p) - - -- -- -------- ---------- -------- ----------------------- T T F F T T T T T F F T F T T T F T T F T F F T F F T T T T T T </pre> Column 5 contains the truth table values for the implies statement. Column 6 contains the truth table values for the inverse of the implies statement. Column 7 contains the truth table values for the converse of the implies statement. Column 8 contains the truth table values for the equivalency statement involving the inverse of the implies statement and the converse of the implies statement. Since the truth table values for column 8 are all T for every possible condition in the truth table, then the statements involved in column 8 are equivalent. Those statements are from Column 6 and Column 7. You can immediately see from Columns 6 and Columns 7 that the two statements have the same truth table values for all possible conditions in the truth table. This is confirmed by the truth table values in column 8. The law of equivalency states: If the truth table values for A and B agree, then the equivalency statement for A and B is true. If the truth table values for A and B disagree, then the equivalency statement for A and B is false. As long as A and B have the same truth values for all possible conditions in the truth table, then the two statements are considered to be equivalent. If they differ for at least one possible condition, then they are not equivalent. In this discussion, A is equal to (~p -> ~q) which is the inverse of the implies statement, and B is equal to (q -> p) which is the converse of the implies statement. The proof has been shown two ways: <pre> 1. The converse of the implied statement is the contra-positive of the inverse of the implies statement. They are therefore equivalent by the law of contra-position. 2. The converse of the implied statement and the inverse of the implied statement are shown to be equivalent by the law of equivalency. </pre> All possible conditions in the truth table are dependent on the number of variables used. Since there were 2 variables used, then there are 2 * 2 = 4 possible conditions in the truth table involving all 2 variables. If there were 3 variables used, then there would be 2 * 2 * 2 = 8 possible consitions in the truth table involving all 3 variables. The truth table for the implies statement itself depends on the following logic: If p is true and q is false then (p -> q) is false. If p is true and q is true then (p -> q) is true. if p is false, then (p -> q) is true. In other words, the implies statement is always true except when p is true and q is false.
