# Question: definition 1 an integer n is a perfect square provided...

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Definition 1.

An integer n is a perfect square provided that the exists a non-negative k such that n = k^2. This

definition can be written in symbolic form using appropriate quantifiers as follows:

Definition 2.

A positive integer n greater than 1 is prime if its only positive factors are 1 and itself.

For the universe of all integers, let O (x); D(x); P(x), and S(x) be the following sentences.

O(x) :x is odd

D(x) :x is divisible by 7

P(x) :x is prime

S(x) :x is a perfect square

Problem 1.

For each of the following statements.

(1) Write the statement in symbolic form.

(2) Determine whether the statement is true or false. If it is false, provide a counter-example.

(3) Write the negation of the statement in symbolic form in which the negation symbol is used in front of a

conditional.

(4) Write a useful negation of the statement as an English sentence that does not use symbols as quantifiers.

(a) For all integers n, if n is not a perfect square, then 2^n-1 is a prime number.

(b) For all integers n, if n is odd, then n is divisible by 7 or n is prime.

Problem 2.

For each of the following statements.

(1) Write the statement in symbolic form.

(2) Determine whether the statement is true or false. If it is false, provide a counter-example.

(3) Write the negation of the statement in symbolic form in which the negation symbol is used in front of a

conditional.

(4) Write a useful negation of the statement as an English sentence that does not use symbols as quantifiers.

(a) For all integers m and n , if m and n are perfect squares, then n-m is even.

(b) There exists an integer m such that m +n is even for all integers nn.