Question
Download Solution PDFLet * be binary operation defined on R by \(\rm p * q=\frac{p+q}{2}, \forall \) p, q ∈ R. The operation is:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
Let * be binary operation defined on R by \(\rm p * q=\frac{p+q}{2}, \forall \) p, q ∈ R
Concept:
If \(\rm p*q = q*p\) then relation is commutative.
If \(\rm (p*q)*r = p*(q*r)\) then relation is associative .
Calculation:
Let * be binary operation defined on R by \(\rm p * q=\frac{p+q}{2}, \forall \) p, q ∈ R
Commutative:
Let p, q ∈ R
Now,
\(\rm p*q=\frac{p+q}{2}=\frac{q+p}{2}=q*p\)
Hence R is commutative.
Associative :
Let p, q, r ∈ R then
\(\rm (p*q)*r=\frac{p+q}{2}*r=\frac{p+q+2r}{4}\)
\(\rm p*(q*r)=p*\frac{q+r}{2}=\frac{2p+q+r}{4}\)
then \(\rm (p*q)*r\ne p*(q*r)\)
Hence R is not associative.
Hence the option (1) correct .
Last updated on Jun 19, 2025
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