a Determine if the given statement is true or false b if the

a. Determine if the given statement is true or false.

b. if the statement is a theorem, determine if the proof? is valid. If now correct the proof or construct a new, valid proof.

c. If the given statemtn is false, modify the statment to create a new statement which is true and also give a valid proof of the new statement.

16. Statement. Theorem? Proof? statement: Let m be an integer. Then m is even m2 is even. Proof? suppose m is an even integer. Then m 2k for some integer k. Therefore, 2(2k2). Since 2k is an integer, m is (2k)2 even. hat is if m is even, then m is even. 2e 1 for some integer (s Suppose m is an odd integer. Then m t Hence, m (2e 1)2 4 4e 2 2e) 1. Since (2 2e) is Thus, if m is odd, then m is odd. And the an inte is odd. is even contrapositive is if m2 is even, then m

Solution

a) Given statement is true

b) Given proofs are valid

First proof is by contrapositive

second proof is by contradiction

both the proofs are valid