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

By Steven R. Lay

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E) For every positive integer n, n2 + n + 41 is prime. ( f ) Every prime is an odd number. (g) No integer greater than 100 is prime. (h ) 3n + 2 is prime for all positive integers n. ( i ) For every integer n > 3, 3n is divisible by 6. ( j ) If x and y are unequal positive integers and xy is a perfect square, then x and y are perfect squares. (k) For every real number x, there exists a real number y such that xy = 2. ( l ) The reciprocal of a real number x ≥ 1 is a real number y such that 0 < y < 1.

We have already seen the need for set notation in describing the context in which quantified statements are understood to apply. It is not our intention to develop set theory in a formal axiomatic way, but rather to discuss informally those aspects of set theory that are relevant to the study of analysis. In Section 1 we establish the basic notation for working with sets, and in the following two sections we apply this to the development of relations and functions. In Section 4 we compare the size of sets, giving special attention to infinite sets.

A) (A ∪ B) ∪ (U \A) (b) (A ∩ B) ∩ (U \A) (c) A ∩ [B ∪ (U \A)] (d) A ∪ [B ∩ (U \A)] (e) (A ∪ B) ∩ [A ∪ (U \B)] (f ) (A ∩ B) ∪ [A ∩ (U \B)] 7. Let A and B be subsets of a universal set U. Define the symmetric difference A B by A B = (A \B) ∪ (B \A). (a) Draw a Venn diagram for A B. (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.