# For Each Binary Operation * Defined Below, Determine Whether * is Commutative Or Associative. On Q, Define A * B = Ab/2 - Mathematics

For each binary operation * defined below, determine whether * is commutative or associative.

On Q, define a * b  = (ab)/2

#### Solution

On Q, * is defined by * b  = (ab)/2

It is known that:

ab = ba &mnForE; a, b ∈ Q

⇒"ab"/2 = "ba"/2 &mnForE; a, b ∈ Q

⇒ * b = * a &mnForE; a, b ∈ Q

Therefore, the operation * is commutative.

For all a, b, c ∈ Q, we have:

(a*b)*c = ("ab"/2) * c = (("ab"/2)c)/2 = (abc)/4

a * (b*c) = a*("bc"/2) = (a("bc"/2))/2 = "abc"/4

∴(a * b) * c = a * (b * c)`

Therefore, the operation * is associative.

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#### APPEARS IN

NCERT Class 12 Maths
Chapter 1 Relations and Functions
Q 2.3 | Page 24