Prove that if A and B are sets such that ..... (elementary algebra)

Question

Prove that if A and B are sets such that AB≠(alpha), then A∪C≠(alpha) for any set C.

Comments

I looked at this earlier and had a go at answering it, but failed miserably!

Answer 1

Problem 1 True or false? (a) [2, 5] = {2, 3, 4, 5} (b) (6, 9] ⊆ [6, 10) (c) {{∅}} ⊆ {∅, {∅}} (d) {1, 2} ∈ {{1, 2, 3}, {1, 3}, 1, 2} (e) {{4}} ⊆ {1, 2, 3, {4}}. Solution: (a) False, 2.5 ∈ [2, 5] but 2.5 6∈ {2, 3, 4, 5}. (b) This is true, since if 6 < x ≤ 9, then 6 ≤ x < 10. (c) This is true since {∅} ∈ {∅, {∅}}. (d) False, since the set {1, 2} is not an element of the set {{1, 2, 3}, {1, 3}, 1, 2}. (e) This is true, since {4} ∈ {1, 2, 3, {4}}. ¥ Problem 2 Give an example, if there is one, of sets A, B and C such that the following are true. If there is no example write “Not possible”. (a) A ⊆ B, B ⊆ C and C ⊆ A (b) A ⊆ B, B 6⊆ C and A 6⊆ C. Solution: (a) Let A = B = C = ∅. (b) Let A = B = {∅} and C = ∅. ¥ Problem 3 Write the power set P(X) for each of the sets (a) X = {S, {S}} (b) X = {1, {2, {3}}}. Solution: (a) P(X) = {∅, {S}, {{S}}, {S, {S}}}. (b) P(X) = {∅, {1}, {{2, {3}}}, {1, {2, {3}}}}.