snhu MAT 230 EXAM ONE This document is proprietary to Southern New Hampshire University. It and the problems within may not be posted on any non-SNHU website. Sonya Hammonds 1
snhu Directions: Type your solutions into this document and be sure to show all steps for arriving at your solution. Just giving a final number may not receive full credit. PROBLEM 1 (a) The domain for all variables in the expressions below is the set of real numbers. Determine whether each statement is true or false. (i) Va By (x + y ? 0) The statement is true. As long as y is greater than x then the sum will be equal to or higher than zero. (ii) 3x Vy (x . y > 0) The statement is false as long as y is zero. (b) Translate each of the following English statements into logical expressions. (i) There are two numbers whose ratio is less than 1. ExBy(x/y <1) (ii) The reciprocal of every positive number is also positive. Vx(x>0-> 1/x > 0)
snhu PROBLEM 2 Prove the following using the specified technique: (a) Let x and y be two real numbers such that x + y is rational. Prove by contrapositive that if x is irrational, then x - y is irrational. First we will assume that x-y is rational. x - y = a/b for some integer where a and b ¥ 0. Since x+y is rational, then the some thereof must be rational, so y must be rational. y = c/d for some integer where c and d ¥ 0. x- c/d = a/b x= a/b+ c/d x = ad + bc/bd Therefor x is rational so x-y must be rational, proving that if x is irrational the x-y would be irrational. (b) Prove by contradiction that for any positive two real numbers, x and y, if x . y ? 50, then either x < 8 or y < 8. Let us assume that x ? 8 and that y ? 8 8*8<50 64 < 50 is a contradiction proving the statement true.