Advanced High-School Mathematics

216 CHAPTER 4 Abstract Algebra

If the binary operation has an identitye, then this identity isunique.
Indeed, ife′were another identity, then we would have

e ︸ =︷︷ ︸

becausee′

is an identity

e∗e′ ︸ =︷︷ ︸

becausee

is an identity

e′.

(Cute, huh?)

Finally, assume that the binary operation ∗ is associative and has
an identity elemente. The elements′∈Sis said to be aninverseof
s∈S relative to∗if s′∗s=s∗s′=e. Note that ifshas an inverse,
thenthis inverse is unique. Indeed, suppose thats′ands′′are both
inverses ofs. Watch this:

s′ = s′∗e = ︸s′∗(s∗s′′) = (︷︷ s′∗s)∗s′′︸

note how associativity is used

= e∗s′′ = s′′.

Exercises

1. In each case below, a binary operation∗ is given on the setZof
   integers. Determine whether the operation is associative, commu-
   tative, and whether an identity exists.

(a)x∗y=x+xy

(b)x∗y=x

(c)x∗y= 4x+ 5y

(d)x∗y=x+xy+y

(e)x∗y=x^2 −y^2

2. Give an example of three vectorsu,v, andwin 3-space such that
   u×(v×w) 6 = (u×v)×w.


3. LetAbe a set and let 2A be it power set. Determine whether the
   operations∩,∪,and + are associative, commutative, and whether
   an identity element exists for the operation.