Problem 15 of Supplementary Exercises of Chapter 1

Let P(m,n) be the statement "m divides n" where the domain of both variables is positive numbers. The truth values of the following statements are:

a) P(4,5) n = km 5 = k * 4 The truth value is false because k is not an integer in this solution

b) P(2,4) n = km 4 = k * 2 2 = k The truth value is true because k is an integer.

c) VmVn P(m,n) The statement says that For every m and n, m divides n. From the first problem we see that there is cases where values of m and n do not make the statement "m divides n" true. So the truth value of the statement is False.

d) 3mVn P(m,n) The statement says that there is a m which for every n satisfies "m divides n" n = km n = k*1 The truth value for the statement is true because there does exist an m that satisfies "m divides n" for any n.

e) 3nVm P(m,n) The statement says that there is a n for which every m satisfies "m divides n" The statement is false because there is not a n that makes "m divides n" always true

f) VnP(1,n) The statement says that for every n and when m is 1 "m divides n" is satisfied. This statement is true because n/1 is always and integer

***NOTE V is the Universal Quantifier and 3 is used as the Existential Quantifier***