Deck 3: The Logic of Quantified Statements

Question 1

Rewrite the following statement in the form if then (where each of the second two blanks are sentences involving the variable x )
Every valid argument with true premises has a true conclusion.

Answer 1

valid argument x, if x has true premises, then x has a true conclusion.

Question 2

Answer 2

a. Given any real number, there is a real number that is less than the given number.
Or: There is no smallest real number.
b. There is a real number that is less than every real number.

Question 3

Rewrite the following statement formally. Use variables and include both quantifiers
in your answer.
Every rational number can be written as a ratio of some two integers.

Answer 3

Question 4

Which of the following is a negation for "There exists a real number x such that for all real
Numbers y, xy > y."

A) There exists a real number x such that for all real numbers
B) There exists a real number y such that for all real numbers
C) There exist real numbers x and y such that
D) For all real numbers x there exists a real number y such that
E) For all real numbers y there exists a real number x such that
F) For all real numbers x and

Question 5

Is the following argument valid or invalid? Justify your answer.
All real numbers have nonnegative squares.
The number i has a negative square.
Therefore, the number i is not a real number.

Question 6

Rewrite the following statement formally. Use variables and include both quantifiers
in your answer.
Every even integer greater than 2 can be written as a sum of two prime numbers.

Question 7

Consider the statement "The square of any odd integer is odd."

Question 8

Question 9

Write negations for each of the following statements:

Question 10

Let T be the statement
(a) Write the converse of T.
(b) Write the contrapositive of T.

Question 11

Are the following two statements logically equivalent? Justify your answer.
(a) A real number is less than 1 only if its reciprocal is greater than 1.
(b) Having a reciprocal greater than 1 is a sufficient condition for a real number to be less
than 1.

Question 12

Which of the following is a negation for "For any integer n, if n is composite, then n is even
Or n > 2."

A) For any integer n , if n is composite, then n is not even or
B) For any integer n , if n is not composite, then n is not even or
C) For any integer n , if n is not composite, then n is not even and
D) For any integer n , if n is not composite, then n is even and
E) For any integer n , if n is not composite, then n is not even and
F) There exists an integer n such that if n is composite, then n is not even and
G) There exists an integer n such that n is composite and n is not even and
H) There exists an integer n such that if n is not composite, then n is not even and
I) There exists an integer n such that n is composite and n is even and
J) There exists an integer n such that if n is not composite, then n is not even or

Question 13

Is the following argument valid or invalid? Justify your answer.
All prime numbers greater than 2 are odd.
The number a is not prime.
Therefore, the number a is not odd.

Question 14

Which of the following is a negation for "For all real numbers r , there exists a number s such that

A) There exists a real number r such that for all real numbers
B) For all real numbers r , there does not exist a number s such that
C) There exists real numbers r and s such that
D) For all real numbers r and
E) There exists a real number r and there does not exist a real number s such that
F) For all real numbers r , there exists a number s such that
G) There exists a real number r such that there does not exist a real number s with

Question 15

Rewrite the following statement in if-then form without using the word "only": A graph with
n vertices is a tree only if it has n − 1 edges.

Question 16

Which of the following is a negation for "Given any real numbers a and b , if a and b are rational then a / b is rational."

A) There exist real numbers a and b such that a and b are not rational and a / b is not rational.
B) Given any real numbers a and b , if a and b are not rational then a / b is not rational.
C) There exist real numbers a and b such that a and b are not rational and a / b is rational.
D) Given any real numbers a and b , if a and b are rational then a / b is not rational.
E) There exist real numbers a and b such that a and b are rational and a / b is not rational.
F) Given any real numbers a and b , if a and b are not rational then a / b is rational.