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MAT 230 EXAM ONE
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Jaime Rowland
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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)fiExA(I)
For all values of n there exists at least one 'y'such that x plus y 0 is TRUE
(O<f.x)fAxE(n)
There exists at least one 'x' for all the values of y such that x times y > 0. If y equals 0 then x is 1/0 which is not real. The is FALSE
(b) Translate each of the following English statements into logical expressions. (i) There are two numbers whose ratio is less than 1.
There are 2 numbers whose ratio is less than 1 which are 3x3y(x/y < 1)
Thereciprocalof everypositivenumberisalsopositive.
ThereciprocalofeverypositivenumberisalsopostiveisVx(x > 0 => 1/x > 0)
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PROBLEM 2 Prove the following using the specified technique: () Let x and y be two real numbers such that x + y is rational. Prove by contrapositive that if x is irrational. then c - u is irrational
The predicate of the whole argument is if x+y is rational and x-y is reasonable, then x is reasonable as well To see the sum of the two logical estimates is often sound. Let us be two real p/9 amount then seeing how. p/q + r/s = (ps+rq)/qs is logical, too. x Thus if x + y is logical and x - y is true, then x + y + x - y = 2x should be sound. Let's assume 2x = p/q Therefore the rational x = p/2q proves the necessary statement in the query.
(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 x times y 50 and x 8 but 8 , both to demonstrate it through contradiction. So, x times y 8 times 8 = 64. This implies that both x times y and 50 and x times y and 64 are valid in the very same period. The argument is vali