I need these three answered so that I have a basis to understand. For each of th

Question

I need these three answered so that I have a basis to understand.

For each of the following statements, (a) write the negation of the statement, in words, as simply as possible, using the appropriate logical equivalences, and (b) prove or disprove the statement. (Be sure to make it clear whether the given statement is true or false.) In either case, make your proof or disproof as general as needed for the given statement. Write all answers as clearly as possible, using complete sentences 1. For al integers n, if 4 divides n and 10 divides n, then 40 divides n 2. For all integers n, if 40 divides n 27, then 4 divides n 3 and 10 divides n +3. . There is an integer a so that for every integer b, 0 dividas ab

Explanation / Answer

Negation of a statement: It is basically the opposite of a statement (this is a very inaccurate statement a is told only to help you understand)

eg:

20 is divisible by 4 (true)

then its negation is:

20 is not divisible by 4 (false)

Answers

1.For all integers n, if 4 divides n and 10 divides n then 40 divides n (false)

This can if rephrased as "if n is an integer that can be divided by 4 and 10 then it is divisible by 40"

To negate this phrase all we have to do is find on integer which does not follow this i.e. the negation statement is :

"there exist an integer that is not divisible by 4 or 10 and is divisible by 40"

Proof for why the statement is false:

if n is divisible by 4 then it can be written of the form

ax2x2 (where a is an integer)

if n is also divisible by 10 then it can be written of the form.

bx5x2 (where b is an integer)

comparing this we fin that one of the '2' could be common and that a number divisible by 4 and 10 is divisible by 20(not necessarily 40) .

STATEMENT2:

"For all integers n, is 40 divides n-27,then 4 divides n-3 and 10 divides n+3"(true)

Negation:

"There exist integer n for which 40 divides n-27 and 4 does not divide n-3 ,or 10 does not divide n+3"

(Another hint: When negated the "and" in a statement turn into "or" and not gets added in the result)

Proof:

Let n- 27 be divisible by 40

=> n-27 = 40b            (b is some integer)

=>n = 40b +27

=> n-3 = 40b + 24

which is divisible by 4.

=>n+3 = 40b +30

which is divisible by 10.

STATEMENT3:

"There is an integer a so that for every integer b, 40 divides ab" (true)

Negation:

"for all integer a there exists an integer b for which 40 does not divide ab" (false)

(Another hint: 'for all' turns to 'there exist' and vv while negating)

Proof:

a=40x implies ab will always be divisible by 40 (b and x are integers)

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