Question

help please and thank you

5. True or False. For each of the following statements, determine whether the statement is True or False and then prove your assertion. That is, for each True statement provide a proof, and for each False statement provide a counterexample (with explanation). Hint: Draw appropriate Venn diagrams to aid your explorations! Let A, B and C be sets (a) A - (B C) (A - B) C (b) (А — В) — С - (А-С) — В. (c) (AU B)- C (A - C)n (B - C) (d) If A B- C, then B AUC (e) If B AUC, then A B- C (f) (A B)UB= A (g) (AUB)-B= A.

Answer 1

Let A, B and a be sets. (a) A (B-C)= (A-B-C False. According to the Venn Diagrams. LA A-(B-C) The Shaded Area. (A-B -C shaded The Area. (6) . (A-B)- c = (A-C)-B True. And this the counter example of (a). E AB CA-B-C shaded Area. (A-2)-B the shaded the Area. (c) = - (AUB)-C (A-2) n (B-C) False . B ca (A-C) n(B-C) The shaded Area, (AUB)-C the shaded Area. The Counter Exarple is (AUB)-C = (A-c) u (B-C) the shaded (A-2) u (B-C) Area

(d) If A=B-C, then B- AUC False. A= Boc the shaded Area. CB]) the non shaded Area ire the shaded Area Is A where fello Avc. I whole Diagrams Counter Example is a which is mot same as Bo A B-C, then B = AUCBnc) The If (e) af B= Ave the n B - C - False B = Auc the shaded Area. The Counter xample If A-B-e WA 3-2 the shaded area. which is not A. is a them A=B-(CA). (f) (A-B) UB = A False. (A- BUB . the shaded Area, which is not equal to A. The Counter Example is - (A-B) U (AnB) = A (A-BUCA nB which the shaded egroal to Area. A. in

| (g) (AUB) - B= A Flare. (AUB)-B the shaded area. which is not equal to A. The Counter example is → . (A-B) U (AnB = A. According to the question (f).