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The binary operation * is defined by on the set Q if all rational numbers. Show that * is associative.

Given that * is a binary operation on Q defined by for all a,b∈Q.


We know that associative property is (p*q)*r = p*(q*r)


Let’s check the associativity of given binary operation:


⇒


⇒


⇒ ...... (1)


⇒


⇒


⇒ ...... (2)


From (1) and (2) we can clearly say that associativity hold for the binary operation ‘*’ on ‘Q’.


More from this chapter

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10

On Z, the set of all integers, a binary operation * is defined by a*b = a + 3b – 4. Prove that * is neither commutative nor associative on Z.

 11

On the set Q of all rational numbers if a binary operation * is defined by , prove that * is associative on Q.

13

On Q, the set of all rational numbers a binary operation * is defined by . Show that * is not associative on Q.

 14

Let S be the set of all rational numbers except 1 and * be defined on S by a*b = a + b – ab, for all a,b∈S.

Prove that:


i. * is a binary operation on S


ii. * is commutative as well as associative.