Question

Show that addition and multiplication are associative binary operation on R. But subtraction is not associative on R. Division is not associative on R∗

Answer 1

Let a, b, c ∈ R

Addition :

( a ∗ b ) ∗ c

= ( a + b ) + c

= a + b + c

And

a ∗ ( b ∗ c )

= a ∗ ( b + c )

= a + b + c

Thus, a ∗ ( b ∗ c ) = ( a ∗ b ) ∗ c

Hence, it is associative.

Multiplication :

( a ∗b ) ∗ c

= ab ∗ c

= abc

And

a ∗ ( b ∗ c )

= a ∗ bc

= abc

Thus, ( a ∗ b ) ∗ c = a ∗ ( b ∗ c )

Hence, it is associative.

Substraction :

( a ∗ b ) ∗ c

= ( a - b ) ∗ c

= a - b - c

And

a ∗ ( b ∗ c )

= a ∗ ( b - c )

= a - b + c

Thus, ( a ∗ b ) ∗ c ≠ a ∗ ( b ∗ c )

Hence, it is not associative.

Division :

( a ∗ b ) ∗ c

= a/b ∗ c

= a/bc

And

a ∗ ( b ∗ c )

= a ∗ b/c

= ac/b

Thus, ( a ∗ b ) ∗ c ≠ a ∗ ( b ∗ c )

Hence, it is not associative

Hence Proved.

Answer 2

Answer:

An operation is commutative if for any a and b, we have ab=ba. Finding one pair a,b such that ab=ba doesn't prove the operation is commutative; this has to hold for every pair.

Consider the set {a,b,c} whose binary operation ⋅ is given by the following:

a⋅a=aa⋅b=ba⋅c=c

b⋅a=bb⋅b=bb⋅c=c

c⋅a=cc⋅b=bc⋅c=a

This operation has a as an identity element. However, it is not commutative (since b⋅c≠c⋅b) and it is not associative (since b⋅(c⋅c)=b≠a=(b⋅c)⋅c).