Question

3) Let S be a set with an associative binary operation :SxS->S. Let e, be a left identity of S (i.e., e, *ssVse S), and let eg be a right identity of S (i.e., a) Prove that e-e b) Also prove that S can have at most one 2-sided identity.

Answer 1

3.

a)

Theorem : The identity element if it exist for any algebraic structure is unique.

Proof:
   Let S be a set with binary operation • : S x S -> S and
   e and e' be left and right identity elements of S respectively.

   ⇒  e, e' ∈ S

   For an associative relation
   
      (e • s) • e' = e • (s • e')    Ɐ s ∈ S       [Given]
   ⇒        s • e' = e • s
   ⇒        s = s 

   Thus,  e = e'

b)
   Here, e and e' were the two identity elements which then condensed to one.
   Thus for every Set S, if identity exists then it must be unique.

Proof: 
   Let, e and e' be two identity elements of S.

   ⇒  e, e' ∈ S

   Let,   e be the identity element
   ⇒  e • e' = e',         ....(1)

   Let,   e' be the identity element
   ⇒  e' • e = e,          ....(2)

   from equation (1) and (2), we get

      e = e'
   Thus, identity element is unique if it exists.

Screenshot:

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