Question

The operation, , is defined on the set of non-negative integers 0,1,2,... by xy = |x-y|.

I) Is it commutative?
a) Yes
b) No

Answer 1

Yes, the operation defined as xy = |x-y| on non-negative integers is commutative because |a-b| = |b-a|, which satisfies the commutative property.

Step-by-step explanation:

The operation defined by xy = |x-y| for non-negative integers is asked to be evaluated for its commutativity. An operation is commutative if the order of its application does not affect the result, which is expressed mathematically as op(a, b) = op(b, a). Let's consider two arbitrary non-negative integers 'a' and 'b'.

In this case, ab = |a-b| and ba = |b-a|. Since the absolute value function |x| satisfies the property that |x| = |-x| for any number 'x', it follows that |a-b| = |b-a|. Thus, ab = ba, proving that the operation is indeed commutative.

Answer: a) Yes