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. Kaitlynn Vogel 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 is true because the statement reads, "For all x, there exists y such that x + y ? 0".Meaningthatwhatevernumberischosenforx, anumber forythatmakesthesumof xandy ? 0canbefound. (ii) 3x Vy (x . y > 0) This statement is false because this statement reads "there is a number x that when you multiply it with any number y, the result is always greater than 0." This is not true because if y is 0, no matter what number x is, x times 0 is always 0. (b) Translate each of the following English statements into logical expressions. (i) There are two numbers whose ratio is less than 1. x <1 y<1 x + y Thisshowsthatthetwonumbers, xandy, arelessthan1.Italsoshowsthatxcannotequaly, becausethentherewouldno Thereciprocalof everypositivenumberisalsopositive. x > 0 1/x > 0 Vx(x>0->1/x>0) Thisreads" that forallnumbersx, wherexisgreaterthan0, thereciprocalof xisalsogreaterthan0."
snhu PROBLEM 2 Prove the following using the specified technique: (ii) 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. To prove something by contrapositive we use "if p then q (p-> q), thecontrapositiveis" if notqthenp" (-q ->> p). So, x - yisrational x+ y =r x - y =S Addingtheseequations, 2x=r+ s x=(r+ s)/2 Provebycontradictionthat foranypositivetworealnumbers,xandy, ifx-y ? 50, then either x < 8 or y < 8. To show proof by contradiction we have to assume the opposite of what we want to prove. So I would assume, either x or y ? 8 x-y ?0 x-y ? 50 Ifxory ? 8, thenx - ycouldbe > 50, whichprovesthecontradiction.