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. Isahiah Stone 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) VxBy (x+y ? 0) The statement says that for all values of x, there exist a value of y such that x plus y is greater than or equal to zero. This statement is true because even with a negative x, as long as the y is greater than x, the sum will be higher zero or higher. (ii) 3x Vy (x . y > 0) The statement says that there exist a x value for all values of y such that x times y is greater than zero. This statement is false if 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. Exay(> < 1) (ii) The reciprocal of every positive number is also positive. V x(x > 0 - 1 > 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. Assume x-y is rational So,x - y = ª for some integers a and b where b# 0 Given that x+y is rational and it is known that the sum of two rational numbers is rational, then y must be rational. So, y"= & for some integers c and d where d # 0 a Plugging in for y, x - a = b x = a +ºc b x = ad+bc bd 01 Therefor x is rational and if x is rational, x-y must be rational. Which proves if x is irrational then x-y is 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. Assume x ? 8 and y ? 8 Plug 8 into the equation, 8 * 8 < 50 64 < 50 is a contradiction which proves the original statement true.