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.
TESTBANKSNERD
1
, 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) ∀ x ∃ y (x + y ≥ 0)
∀x ∃y (x + y ≥ 0)
∀x means “for every real number x” ∃y means there exists some real number y depending
on x. We need to find at least one y for each x such that
x+y ≥0
Expressing y in terms of x
Choose
y = −x
Then
x + y = x + (−x) = 0
This means
0≥ 0
hence the inequality holds. Alternatively, any y ≥ −x which satisfies x + y ≥ 0 Therefore,
since for every real x we can find such a y, the statement is TRUE.
(ii) ∃ x ∀ y (x · y > 0)
∃x ∀y (x · y > 0)
For any x ∈ R, let y = 0. Then
x · y = x · 0 = 0 /> 0
Hence, no x exists that makes x · y > 0 true for all y Therefore, the 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.
Let x and y be numbers. Since there are two numbers, it implies an existential quantifier
and the ratio must be defined (y /= 0), the statement become
∃x ∃y (y /= 0 ∧ x/y < 1)
(ii) The reciprocal of every positive number is also positive.
Let x be a positive number (x > 0). Its reciprocal is 1/x. Since the statement asserts
that this reciprocal is positive (1/x > 0) for all x > 0.
Therefore, the logical expression will be
∀x (x > 0 → 1/x > 0)
This document is proprietary to Southern New Hampshire University. It and the problems within
may not be posted on any non-SNHU website.
TESTBANKSNERD
1
, 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) ∀ x ∃ y (x + y ≥ 0)
∀x ∃y (x + y ≥ 0)
∀x means “for every real number x” ∃y means there exists some real number y depending
on x. We need to find at least one y for each x such that
x+y ≥0
Expressing y in terms of x
Choose
y = −x
Then
x + y = x + (−x) = 0
This means
0≥ 0
hence the inequality holds. Alternatively, any y ≥ −x which satisfies x + y ≥ 0 Therefore,
since for every real x we can find such a y, the statement is TRUE.
(ii) ∃ x ∀ y (x · y > 0)
∃x ∀y (x · y > 0)
For any x ∈ R, let y = 0. Then
x · y = x · 0 = 0 /> 0
Hence, no x exists that makes x · y > 0 true for all y Therefore, the 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.
Let x and y be numbers. Since there are two numbers, it implies an existential quantifier
and the ratio must be defined (y /= 0), the statement become
∃x ∃y (y /= 0 ∧ x/y < 1)
(ii) The reciprocal of every positive number is also positive.
Let x be a positive number (x > 0). Its reciprocal is 1/x. Since the statement asserts
that this reciprocal is positive (1/x > 0) for all x > 0.
Therefore, the logical expression will be
∀x (x > 0 → 1/x > 0)
