MATH SOLVE

3 months ago

Q:
# 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.

Accepted Solution

A:

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 numberb=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=3a-b=8-6=2b-c=6-3=3a-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 numberc=Odd numbera=9,b=6,c=5a-b=9-6=3b-c=6-5=1a-c=9-5=4Here, a-b and b-c are not both even but a-c is even .Hence, the inverse statement is false.