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4 November, 16:32

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.

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  1. 4 November, 16:54
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    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.
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