Question

Consider the binary operations * : R x R → R and o : R x 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 * b). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over? Justify your answer.

Answer 1

∀ a, b ∈ R a * b = |a – b| = |b – a| = b * a 

hence commutative ∀ 3, 5 ,1 ∈ R 

(3 * 5) * 7 |3 – 5| * 7 

= 2 * 7= |2 – 7| = 5 

3* (5 * 7) = 3* |5 – 7| 

= 3 * 2= |3 – 2| = 1 

a* (b * c) & (a * b) *c 

∴ is * not associative. 

a o b = a, b o a = b 

hence a o b ≠ b o a 

hence not commutative 

(a o b)o c : a o c = a 

a o (b o c) = a o b = a 

hence o is associative 

a*(b o c) (a * b)=|a – b| 

(a * b) o (a * c) = |a – b| o |a – c| 

= |a – b| 

hence a* (b o c) = (a * b) o (a * c) 

hence o is distributive over *.