Math

untitled folder/Logic.rtf

1. Define F: P({a, b, c}) → Z as follows: For all A in P({a, b, c}), F(A) = the number of elements in A. a. Is F one-to-one? Prove or give a counterexample. b. Is F onto? Prove or give a counterexample. 2.Let S be the set of all strings in a’s and b’s, and define C: S → S by C(s) = as, for all s ∈ S. (C is called concatenation by a on the left.) a. Is C one-to-one? Prove or give a counterexample. b. Is C onto? Prove or give a counterexample 3.Define H: R °ø R → R °ø R as follows: H(x, y) = (x + 1, 2 − y) for all (x, y) ∈ R °ø R. a. Is H one-to-one? Prove or give a counterexample. b. Is H onto? Prove or give a counterexample 4.Suppose F:X → Y is onto. Prove that for all subsets B ⊆ Y, F(F−1(B)) = B. For 5-6 let X and Y be sets, let A and B be any subsets of X, and let C and D be any subsets of Y . Determine which of the properties are true for all functions F from X to Y and which are false for at least one function F from X to Y . Justify your answers 5.For all subsets A and B of X, F(A − B) = F(A) − F(B). 6.For all subsets C and D of Y , F−1(C ∩ D) = F−1(C) ∩ F−1(D). Show that each of the following holds for all subsets A and B of S and all u ∈ S. a. χA∩B(u) = χA(u) °§χB(u) b. χA∪B(u) = χA(u) + χB(u) − χA(u) °§χB(u)