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We have already used phrases like for all, there exists, and, or, if. . then, etc. What exactly do these mean? 4 change if we switched the order of some of these phrases? We will also need to express the negations of mathematical statements. Think about what the negation of an and statement should look like; for example, what are the negations of the statements “John and Mary like cookies” or “there exist positive integers that are not prime”? M. Beck and R. 1 Quantifiers The words for any usually means ∀, but for any is sometimes used (misused in our opinion) to mean ∃.

32 it is not clear that S has a smallest element. 34. If m and n are integers that are not both 0, then S = {k ∈ N : k = mx + ny for some x, y ∈ Z} is not empty. 35. Compute gcd(4, 6), gcd(7, 13), gcd(−4, 10), and gcd(−5, −15). 36. Given a nonzero integer n, compute gcd(0, n) and gcd(1, n). Review Question. Are you able to use the method of induction correctly? Weekly reminder: Reading mathematics is not like reading novels or history. You need to think slowly about every sentence. Usually, you will need to reread the same material later, often more than one rereading.

Iii) D = B. 4. Let A, B,C be sets. (i) A = A. (ii) If A = B then B = A. (iii) If A = B and B = C then A = C. 1 when we talked about equality of two integers. We called the properties reflexivity, symmetry, and transitivity, respectively. 1. 5. When reading or writing a set definition, pay attention to what is a variable inside the set definition and what is not a variable. As examples, how do the following pairs of sets differ? (i) S := {m : m ∈ N} and Tm := {m} for a specified m ∈ N. (ii) U := {my : y ∈ Z, m ∈ N, my > 0} and Vm := {my : y ∈ Z, my > 0} for a specified m ∈ N.

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