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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.
Solution
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.
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