Consider finite sets A, B, C. For the following statements, prove it if it is true,...

 

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Consider finite sets A, B, C. For the following statements, prove it if it is true, or disprove it if it is false. (a) (4 pts) AU (BOC) = (AUB) n(AUC). (Hint: First prove "if z EAU (BOC), then TE (AUB) n(AUC)". Then prove "if a € (AUB) n(AUC), then x EAU (BNC)".) (b) (4 pts) An B|≤|A|-|B. (c) (4 pts) A\B|2|A|-|B|. (d) (4 pts) |A|+|B| = |AU B| if and only if A and B are disjoint. (e) (4 pts) AUBUC| = |A|+|B|+|C|-|AnB|-|AnC|-|BnC|+|AnBnC). (f) (4 pts) If f: A → B is a bijection, and g: B → C is a bijection, then h: A → C defined as h(x) = g(f(x)) is a bijection. (g) (4 pts) Consider any functions f: A → B and g: B → C (which may or may not be bijections). If h: A→C defined as h(x) = g(f(x)) is a bijection, then both f and g must be bijections. Consider finite sets A, B, C. For the following statements, prove it if it is true, or disprove it if it is false. (a) (4 pts) AU (BOC) = (AUB) n(AUC). (Hint: First prove "if z EAU (BOC), then TE (AUB) n(AUC)". Then prove "if a € (AUB) n(AUC), then x EAU (BNC)".) (b) (4 pts) An B|≤|A|-|B. (c) (4 pts) A\B|2|A|-|B|. (d) (4 pts) |A|+|B| = |AU B| if and only if A and B are disjoint. (e) (4 pts) AUBUC| = |A|+|B|+|C|-|AnB|-|AnC|-|BnC|+|AnBnC). (f) (4 pts) If f: A → B is a bijection, and g: B → C is a bijection, then h: A → C defined as h(x) = g(f(x)) is a bijection. (g) (4 pts) Consider any functions f: A → B and g: B → C (which may or may not be bijections). If h: A→C defined as h(x) = g(f(x)) is a bijection, then both f and g must be bijections.