Provide counterexamples to the following statements If n2 0

Provide counterexamples to the following statements If n^2 > 0, then n > 0. If n is an even number, then n^2 + 1 is prime. If a real number is not positive, then it must be negative. The number n is an even integer if and only if 3n+2 is an even integer. All people with red hair have green eyes and are tall.

Solution

a)

False

Counterexample:

n=-1 , n^2=1>0

b.

False

Counterexample

n=8,

n^2+1=8^2+1=65=13*5

c)

False

0 is not positive but also not negative

d)

True

If n is even then 3n is even number as any multiple of even number is even

Hence, 3n+2 is also even

Now let 3n+2 be an even number.

Hence, 3n is an even number

3 is odd number hence n must be even number

e)

False

People with red hair can be short also