Consider the following statement: ∀ a, b, c ∈ Z, if a - b is even and b - c is even, then a - c is even.

Write the converse and inverse of this statement. Indicate (by formal reasoning) which among the statement, its converse and its inverse, are true and which are false. Give a counterexample for each that is false.

Answer Step-by-step explanation:

Given statement if a-b is even and b-c is even then a-c is even.

Let p: a-b and b-c are even

q: a-c is even.

Converse: If a-c is even then a-b and b-c are both even.

Inverse:If a-b and b-c are not both even then a-c is not even.

If a = Even number

b = Even number

c=Even number

If a-c is even then a-b and b-c are both even ... Hence, the converse statement is true.

If a=Odd number

b=Odd number

c = Odd number

If a-c is even then a-b and b-c are both even number. Hence, the converse statement is true.

If a=Even number

b = Even number

c = Odd number

a-b and b-c are both odd not even number but a-c is even number

a=8, b=6 c=3

a-b=8-6=2

b-c=6-3=3

a-c=8-3=5

If a-c is odd then a-b even but b-c is odd. Hence, the converse statement is false. But the inverse statement is true.

If a = Odd number

b=Even number

c = Even number

If a-b is odd and b-c is even then a-c is odd not even. Hence, the inverse statement is true.

If a = Odd number

b=Eve number

c=Odd number

a=9, b=6, c=5

a-b=9-6=3

b-c=6-5=1

a-c=9-5=4

Here, a-b and b-c are not both even but a-c is even. Hence, the inverse statement is false.