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Deck 3: The Logic of Quantified Statements
ملء الشاشة (f)
سؤال
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.
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.
سؤال
سؤال
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.
in your answer.
Every rational number can be written as a ratio of some two integers.
سؤال
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
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
سؤال
سؤال
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.
in your answer.
Every even integer greater than 2 can be written as a sum of two prime numbers.
سؤال
Consider the statement "The square of any odd integer is odd." 
سؤال
سؤال
Write negations for each of the following statements: 
سؤال
Let T be the statement 
(a) Write the converse of T.
(b) Write the contrapositive of T.
(a) Write the converse of T.
(b) Write the contrapositive of T.
سؤال
سؤال
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
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
سؤال
سؤال
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
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
سؤال
سؤال
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العب
ملء الشاشة (f)
Deck 3: The Logic of Quantified Statements
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.
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.
valid argument x, if x has true premises, then x has a true conclusion. 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.
Or: There is no smallest real number.
b. There is a real number that is less than every real number.
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.
in your answer.
Every rational number can be written as a ratio of some two integers.
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
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
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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.
All real numbers have nonnegative squares.
The number i has a negative square.
Therefore, the number i is not a real number.
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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.
in your answer.
Every even integer greater than 2 can be written as a sum of two prime numbers.
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7
Consider the statement "The square of any odd integer is odd."
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8
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9
Write negations for each of the following statements:
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10
Let T be the statement
(a) Write the converse of T.
(b) Write the contrapositive of T.
(a) Write the converse of T.
(b) Write the contrapositive of T.
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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.
(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.
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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
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
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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.
All prime numbers greater than 2 are odd.
The number a is not prime.
Therefore, the number a is not odd.
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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
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
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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.
n vertices is a tree only if it has n − 1 edges.
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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.
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.
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