Let A, B, C be three sets. Show that A ∪ B = A ∩ C ⇐⇒ B ⊆ A ⊆ C.
(5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C) (5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C)
Let A and B be two sets. (a) Show that Ac = (Ac ∩ B) ∪ (Ac ∩ Bc ), Bc = (A ∩ Bc ) ∪ (Ac ∩ Bc ). (b) Show that (A ∩ B) c = (Ac ∩ B) ∪ (Ac ∩ Bc ) ∪ (A ∩ Bc ). (c) Consider rolling a fair six-sided die. Let A be the set of outcomes where the roll is an odd number. Let B be the set of outcomes...
DISCRETE/LOGIC MATH please show work and explain 3. Let A, B, C be sets. Use the Venn Diagram below to help give a counterexample to the statements in parts (a) and (b). In each case, if the Venn diagram suggests a relationship between the LHS and RHS (without any additional hypotheses), then state and prove it. с (a) For all sets A,B,C, (A (b) For all sets A, B, C, ( A B B C = A ( B C...
Let A, B and C be sets. Prove
#9-11 please 9. Let A and B be disjoint sets in the universe U. Let C be a proper subset of A. (a) Draw a Venn Diagram representing this information. (b) What is BAC? 10. Let A be a set in the universe U. (a) Draw a Venn Diagram and shade in the region A. Then draw another Venn Diagram with the same set A, but shade in A'. (b) What is A'U A? 11. Give an example of three...
Let A,B and C sets. Assume A ⊂ B ⊂ C, prove C \B = C \(A∪B)
Let A, B, and C be sets. Prove the following statement: (A − B) ∩ (C − A) = ∅
15 Let A and B be contented sets with B negligible. Apply Proposition 2.7 with 4to show that A U B is contented with A u B D(A U B) D(A)(B). 15 Let A and B be contented sets with B negligible. Apply Proposition 2.7 with 4to show that A U B is contented with A u B D(A U B) D(A)(B).
10. Let A, B, and C be sets. (a) Prove or disprove: if A - C CB-C, then ACB. (b) State the converse of part (a) and prove or disprove.
For nonempty sets A, B and C, let f : A → B and g : B → C be functions. Prove that if g ◦ f is injective, then f is injective