For each of the following statements, determine its truth value. If the statemen

Question

For each of the following statements, determine its truth value. If the statement is false, provide a counterexample. Prove at least two of the true statements.

1. For all n N, n2 5.

2. There exists n N such that n^21=0.

3. There exists x N such that for all y N, yx.

4. For all x Z, x3 x.

5. For all n Z,there exists m Z such that n+m=0.

6. There exists integers a and b such that 2a + 7b = 1.

7. There do not exist integers m and n such that 2m + 4n = 7.

8. For all integers a, b, c, if a divides bc, then either a divides b or a divides c.

Explanation / Answer

1.

False.

n=1,n^2=1

2.

For n=1,n^2-1=0.

Hence truth value is: True

3.

False

For any x we can choose ,y=x+1 and we would have :y>x.

4.

False.

Take x to be a negative integer. For example,let x=-2

x^3=-8<-2=x

5.

True.

For any integer n we can let:m=-n which is also an integer and we have:m+n=0

6.

True

This is because gcd(2,7)=1

We can find a,b using the Euclid algorithm.

We can easily see:a=4,b=-1 is a solution:

2*4+7*(-1)=1

7.

True

Because 2m+4n would be an even number and 7 is an odd number

8.

False.

Let a=6, b=2,c=3

a does not divide b,a does not divide c

But bc=6 and a divides bc

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.