# Download Analysis with an introduction to proof by Steven R. Lay PDF

By Steven R. Lay

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B) What is A A? (c) What is A ∅? (d) What is A U ? Sets and Functions 8. Let S = {∅, {∅}}. Determine whether each of the following is True or False. Explain your answers. (a) ∅ ⊆ S (b) ∅ ∈ S (c) {∅} ⊆ S (d) {∅} ∈ S 9. 13(a). THEOREM: Let A be a subset of U. Then A ∪ (U \A) = U. Proof: If x ∈ A ∪ (U \A), then x ∈ __________ or x ∈ __________. Since both A and U \A are subsets of U, in either case we have __________. Thus ____________ ⊆ ____________. On the other hand, suppose that x ∈ ____________.

Intuitively, union may be thought of as putting together, intersection is like cutting down, and complementation corresponds to throwing out. 8 DEFINITION Let A and B be sets. The union of A and B (denoted A ∪ B), the intersection of A and B (denoted A ∩ B), and the complement of B in A (denoted A \ B) are given by A ∪ B = {x : x ∈ A or x ∈ B} A ∩ B = {x : x ∈ A and x ∈ B} A \ B = {x : x ∈ A and x ∉ B}. If A ∩ B = ∅, then A and B are said to be disjoint. 9 PRACTICE iff iff iff (x ∈ A) ∨ (x ∈ B) (x ∈ A) ∧ (x ∈ B) (x ∈ A) ∧ ~ (x ∈ B).

If x ∈ A, then __________ ________________________________________________________. Thus A ⊆ B. ♦ 12. Suppose you are to prove that set A is a subset of set B. Write a reasonable beginning sentence for the proof, and indicate what you would have to show in order to finish the proof. 13. Suppose you are to prove that sets A and B are disjoint. Write a reasonable beginning sentence for the proof, and indicate what you would have to show in order to finish the proof. 14. Which statement(s) below would enable one to conclude that x ∈ A ∪ B?