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MAT 230 EXAM ONE
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Terry Bishop
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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. (0 Z fi+x)fExA(I)
The statements reads, For all values of x, there exists a value of y such that x + y > 0 This statement is True.
(0<f.x)fAxE(m)
The statement reads, There exists a value of x for all values of y such that x * y : 0. If y = 0, then x is 1/0 which is not a real number. This statement is False.
(b) Translate each of the following English statements into logical expressions. (i) There are two numbers whose ratio is less than 1.
The first part of the statement reads There are two numbers, which logically expressed would be: REE The second part of the statement reads whose ratio is less than 1. This would be expressed as: <1 y
Therefore the logical expression would be: 3x 3y(s<1)
(ii) The reciprocal of every positive number is also positive.
Let x be a number. The reciprocal of x is The statement every positive number would be expressed as follows: 0xxA Therefore the logical expression is as follows: Vx(x>0->0)
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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 that x - y is rational. So, x - y = 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 ) Plugging y = intox -- y = f gives the following: =(where d 0 and b 0 (0#q pue 0#poym)5+=x
Therefore x is rational. If x is rational then x - y is rational. Contrapositively if x is rational 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.
If x 8 and y 8. Plugging these into the inequality gives the following: x*y>8*8 x 8 y 64 This is a contradiction to x * y 50. Therefore, it is true that for any positive two real numbers, x and y, if x * y 50, then either