Discrete Mathematical Structures by Pearson

Algebraic Structures 7-49

2. Addition is commutative in R/S

(S + a) + (S + b) = S + (a + b) = S + (b + a)

 = (S + b) + (S + a)

3. Existence of additive identity

We have S = S + 0 ∈ R/S

If S + a ∈ R/S, then (S + 0) + (S + a) = S + (0 + a)

= S + a

∴S is the identity

4. Existence of additive inverse

Let S + a ∈ R/S, then S + (-a) ∈ R/S.

Now, [S + (-a)] + [S + a] = S + [-a + a] = S + 0 = S

Hence, S + (-a), that is, S - a is the additive inverse of S + a.

5. Multiplication is associative

[(S + a)(S + b)](S + c) = (S + ab)(S + c) = S + (ab)c

= S + a(bc) = (S + a)(S + bc) = (S + a)[(S + b)(S + c)]

6. Multiplication is distributive with respect to addition

(S + a)[(S + b) + (S + c)] = (S + a)[S + (b + c)] = S + a(b + c)

= S + (ab + ac) = (S + ab) + (S +