BINARY OPERATIONS: BASIC CONCEPT OF BINARY OPEATIONS

Definition:

Binary operation is any rule of combination of any two elements of a given non empty set. The rule of combination of two elements of a set may give rise to another element which may or not belong to the set under consideration.

It is usually denoted by symbols such as, *, Ө e.t.c.

 

Properties:

A. Closure property: A non- empty set z is closed under a binary operation * if for all a, b € Z.

Example; A binary operation * is defined on the set S= {0, 1, 2, 3, 4} by

X*Y = x + y –xy. Find (a) 2 * 4 (b) 3* 1 (c) 0* 3.  Is the set S closed under the operation *?

Solution

(a)  2 * 4, i.e, x= 2,y=4

2+ 4 – (2×4)       = 6-8 = -2.

(b)  3* 1 = 3+1-( 3x 1)    = 4 – 3= 1

(c)   0*3  = 0 + 3 –( 0 x3) = 3

Since -2€ S, therefore the operation * is not closed in S.

B. Commutative Property: If set S, a non empty set is closed under the binary operation *, for all a,b€ S. Then the operation * is commutative if a*b= b*a

Therefore, a binary operation is commutative if the order of combination does not affect the result.

Example; The operation * on the set R of real numbers is defined by:

p*q= p+ q3-3pq. Is the operation commutative?

 

Solution

p*q= p3 + q3 -3pq

Commutative condition p*q= q*p

To obtain q*p, use the same operation q*p, use the same operation p*q but replace p by q and q by p.

Hence, q*p= p3+ q-3qp

In conclusion p*q= q*p, the operation is commutative.

 

C. Associative Property: If a non – empty set S is closed under a binary operation *, that is a*b €S. Then a binary operation is associative if (a*b) * c= a*(b*c)

Such that C also belongs to S.

Example: The operation Ө on the set Z of integers is defined by; a Ө b = 2a +3b -1. Determine whether or not the operation is associative in Z.

Solution

Introduce another element C

Associative condition: (aӨb) Өc = a Ө (b Өc)

(aӨb)Өc = (2a+ 3b- 1) Ө C

= 2(2a +3b -1) + 3c -1

= 4a + 6b- 2+ 3c- 1

= 4a +6b+3c- 3.

Also, the RHS, a Ө (b Ө c) = a Ө (2b+3c- 1)

= 2a+ 3(2b +3c- 1) – 1

= 2a + 6b +9c -3 -1

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