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Consider the binary operations * : R × R → R and o : R × R → R defined as a * b = |a– b| and a o b =a, ∀ a, b ∈ R. Show that ∗ is commutative but not associative, o is associative but not commutative. Further, show that ∀ a, b, c ∈ R, a * (b o c) = (a * b) o (a* c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.
Consider the binary operations * : R × R → R and o : R × R → R defined as a * b = |a– b| and a o b =a, ∀ a, b ∈ R. Show that ∗ is commutative but not associative, o is associative but not commutative. Further, show that ∀ a, b, c ∈ R, a * (b o c) = (a * b) o (a* c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.
CBSE | Class 12 | Maths | Miscellaneous | NCERT | Relations and Functions
Consider the binary operations * : R × R → R and o : R × R → R defined as a * b = |a– b| and a o b =a, ∀ a, b ∈ R. Show that ∗ is commutative but not associative, o is associative but not commutative. Further, show that ∀ a, b, c ∈ R, a * (b o c) = (a * b) o (a* c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.

Solution:

Stage 1: Check for commutative and cooperative for activity *. a * b = |a – b| and b * a = |b – a| = (a, b)

Activity * is commutative.

a(bc) = a|b-c| = |a-(b-c)| = |a-b+c| and (ab)c = |a-b|c = |a-b-c|

In this way, a(bc) ≠ (ab)c

Activity * is cooperative.

Stage 2: Check for commutative and cooperative for activity o. aob = a ∀ a, b ∈ R and boa = b

This infers aob boa

Activity o isn’t commutative.

Once more, an o (b o c) = an o b = an and (aob)oc = aoc = a Here ao(boc) = (aob)oc

Activity o is cooperative.

Stage 3: Check for the distributive properties

In the event that * is distributive over o, RHS:

= LHS

Also, LHS

RHS

LHS ≠ RHS

Henceforth, activity o doesn’t disperse over.

i

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