Exam 3: The Logic of Quantified Statements

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.

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.

Which of the following is a negation for "For all real numbers r , there exists a number s such that $r s > 10 . "$

For each of the following statements, (1) write the statement informally without using variables or the symbols $\forall$ or $\exists$ , and (2) indicate whether the statement is true or false and briefly justify your answer. (a) $\forall$ real numbers $x , \exists$ a real number $y$ such that $x < y$ . (b) $\exists$ a real number $y$ such that $\forall$ real numbers $x , x < y$ .

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

Consider the statement "The square of any odd integer is odd." (a) Rewrite the statement in the form $\forall$ ___ $n$ ,____. (Do not use the words "if" or "then.") (b) Rewrite the statement in the form $\forall$ ______ $n$ , if ____then_____ (Make sure you use the variable $n$ when you fill in each of the second two blanks.) (c) Write a negation for the statement.

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

For each of the following statements, (1) write the statement informally without using variables or the symbols $\forall$ or $\exists$ , and (2) indicate whether the statement is true or false and briefly justify your answer. (a) $\forall$ integers $a , \exists$ an integer $b$ such that $a + b = 0$ . (b) $\exists$ an integer $a$ such that $\forall$ integers $b , a + b = 0$ .

Write negations for each of the following statements: (a) For all integers $n$ , if $n$ is prime then $n$ is odd. (b) $\forall$ real numbers $x$ , if $x < 1$ then $\frac { 1 } { x } > 1$ . (c) For all integers $a$ and $b$ , if $a ^ { 2 }$ divides $b ^ { 2 }$ then $a$ divides $b$ . (d) $\forall$ real numbers $x$ , if $x ( x - 2 ) > 0$ then $x > 2$ or $x < 0$ . (e) $\forall$ real numbers $x$ , if $x ( x - 2 ) \leq 0$ then $0 \leq x \leq 2$ . (f) For all real numbers $x$ and $y$ with $x < y$ , there exists an integer $n$ such that $x \leq n \leq y$ .

Rewrite the following statement in the form $\forall$ _______x, 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.

Let T be the statement $\forall \text { real numbers } x \text {, if } - 1 < x \leq 0 \text { then } x + 1 > 0 \text {. }$ (a) Write the converse of T. (b) Write the contrapositive of T.

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.

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."

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

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

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.