Question

Define an operation * on Q as follows: a * b = `(("a" + "b")/2)`; a, b ∈ Q. Examine the closure, commutative and associate properties satisfied by * on Q.

Answer 1

Given a * b = `(("a" + "b")/2)`; a, b ∈ Q

a ∈ Q and b ∈ Q

⇒ a * b = `("a" + "b")/2` ∈ Q

Hence * is a binary operation on Q

a * b = `("a" + "b")/2`

b * a = `("b" + "a")/2`

= `("a" + "b")/2` .......[∵ a + b = b + a]

∴ Binary operation * is commutative

a * (b * c) = a * `(("b" + "c")/2)`

= `("a" + ("b" + "c")/2)/2`

= `(2"a" + "b" + "c")/2`

(a * b) * c = `(("a" + "b")/2)` * c

= `("a" + "b" + 2"c")/4`

So, a * (b * c) ≠ (a * b) * c

Hence, the binary operation * is not associative.