Question

Give a proof or counterexample, whichever is appropriate.

1. For any sets A and B,

(A ∩ B = ∅) AND (A ∪ B = B) ⇒ A = ∅

2. An integer n is even if n2 + 1 is odd.
3. The converse of the assertion in exercise 62 is false.
4. For all integers n, the integer n2 + 5n + 7 must be positive. 1.65. For all integers n, the integer n4 + 2n2 − 2n cannot be negative.

Answer 1

. Give a proof or counter example, which enor is appropiate. 3. For any set A and B AnB = cp and AUB=B then A= 0. Proof: suppose A& . then there existan. element in A, let xt A then XE AUB. given that AUB-Bo y FB. so we have ne A and XEB. at ang but it's ahogiven that AnB=0 So neant. A must be empty . A-0 (proved) 2. An integer is even if t1 s odd. This statement in troue. Prot suppose n²+1 is odd. So n² is even. [because for two consicutine integers one in odd another agan na nxn is even beren) we know that product of two integers is even then one of them must be enen. here both no are egalal, so, we say if n2 is even a ny even (prowed)

4 for all integen on must be positive. the integer ntsh to (troue). Prati ²4 5047 = 42. - 2 + 0 +7 m+/2) +- 25 (+52) + since (n +32) is positive. Sosn + 7 = (1 + 7/27 + ²70. a nagnait is & must be positive. (proved) 5 for all integer or, the integens na an2-27 can't be negative. this ĥ trued shoot of 2n²-27 = 3 + 2n =) We know that n2 n for any net. > 2027 an = 212 - 2017 0 y n4+ 2n²-any na Ent - 294 2n²-2n7o . ntt 20²2h is always positine, can't 7920 alareys