Page 28 - DMTH403_ABSTRACT_ALGEBRA

Unit 2: Groups

2.1 Binary Operations Notes

As you all know common operations of addition and multiplication in R, Q and C. All these
operations are examples of binary operations. It can be defined as:
Definition: Let S be a non-empty set. Any function * : S × S  S is called a binary operation
on S.
So, a binary operation associates a unique element of S to every ordered pair of elements of S.
For a binary operation * on S and (a, b)  S × S, we denote *(a, b) by a*b.
We will use symbols like +, –, ×, , o, * and A to denote binary operations.

Let us look at some examples.
(i) + and x are binary operations on Z. In fact, we have + (a, b)= a + b and × (a, b) = a × b  a,
b  Z. We will normally denote a × b by ab.
(ii) Let (S) be the set of all subsets of S. Then the operations  and  are binary operations
on (S), since A  B and A  B are in (S) for all subsets A and B of S.
(iii) Let X be a non-empty set and F(X) be the family of all functions f : X X. Then the
composition of functions is a binary operation on F(X), since fog  F(X)  f, g  F(X).
After defining a non-empty set lets define properties of binary operations.

Definition: Let * be a binary operation on a set S. We say that
(i) * is closed on a subset T of S, if a * b  T  a, b  T
(ii) * is associative if, for all a, b, c  S, (a * b) * c = a * (b * c).

(iii) * is commutative if, for all a, b  S, a * b = b * a.
For example, the operations of addition and multiplication on R are commutative as well as
associative. But, subtraction is neither commutative nor associative on R. Why? Is a – b = b – a or
(a – b) – c = a – (b – c) 4) a, b, c  R ? No, for example, 1 – 2 ! 2 – 1, and (1 – 2) – 3 1 – (2 – 3). Also
subtraction is not closed on N  R. because 1  N. 2  N but 1 – 2  N.

Note A binary operation on S is always closed on S, but may not be closed on a subset
of S.

Task For the following binary operations defined on R, determine whether they are
commutative or associative. Are they closed on N?
(a) x  y = x + y – 5
(b) x * y = 2(x + y)
x y

(c) x  y =
2
for all x, y  R.

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