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. Enter your name here 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) Vx By (x + y > 0) This statement says that for every number x there is a number y such that the sum of x and y is more than or equal to zero. This statement is true. even if the number x is negative, as long as the absolute value of y is greater than the absolute value of x, their sum will be greater than 0. (ii) Ex Vy (x . y > 0) This statement says that for a specific number x and every number y, the product of x and y is more than zero. This is false because if y is 0, the product of x and y is 0. (b) Translate each of the following English statements into logical expressions. (i) There are two numbers whose ratio is less than 1. Axây ( < 1) (ii) The reciprocal of every positive number is also positive. Vx (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. Proof: Assume x - y is rational. We will be proving that x is rational. Since x + y is also rational, the sum of the two expressions is also rational: (x-y)+(x+y) = 2x Therefore, 2x is rational since it is the sum of the two rational expressions. Since 2 is a rational number, x must be a rational number because their product is rational. Therefore, by contrapositive, we have proven that if x is irrational, x - y is irrational. END OF PROOF. (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. Proof: Suppose that for any positive two real numbers x and y, if x . y ? 50, then x ? 8 and y ? 8. Assume x = 8 and y = 8. When the numbers are plugged into x · y ? 50: 8.8 ?50 64?50 This shows an inconsistency, which proves that if x . y ? 50, then either x < 8 or y < 8. END OF PROOF.