BINARY OPERATIONS: BASIC CONCEPT OF BINARY OPEATIONS

CONTENT

Concept of binary operations,
Closure property
Commutative property
Associative property and
Distributive property.

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:

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

2 * 4, i.e, x= 2,y=4

2+ 4 – (2×4) = 6-8 = -2.

3* 1 = 3+1-( 3x 1) = 4 – 3= 1
0*3 = 0 + 3 –( 0 x3) = 3

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

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= p³+ 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+ q³-3qp

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

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

a Ө (b Ө c) = 2a+ 6b+ 9c -4

Since, (a Ө b) Ө c ≠ a Ө (b Ө c), the operation is not associative in Z.

Evaluation

An operation* defined on the set R of real numbers is

x* y = 3x+ 2y- 1, x,y €R. Determine (a) 2*3 (b) -4* 5 (c) 1 * 1

3 2 is the operation closed.

Distributive Property:

If a set is closed under two or more binary operations (* Ө) for all a, b and c € S, such that:

a*(bӨ c) = (a*b )Ө( a*c – Left distributive

(BӨc) *a = (b*a) Ө(c*a) – Right distributive over the operation Ө

Example: Given the set R of real numbers under the operations * and Ө defined by:

a*b = a+ b- 3, aӨb= 5ab for all a, b € R. Does * distribute over Ө.

Solution Let a, b,c € R

a* ( bӨc) = (a*b) Ө (a*c)

a* (bӨc) = a* (5ab)

= a+ 5ab -3.

(a*b) Ө (a*c) = (a+ b -3) Ө ( a+ c-3)

= 5(a +b-3)(a +c -3)

From the expansion, it’s obvious that, a* ( bӨc) ≠ (a*b) Ө (a*c) therefore * does not distribute over Ө.

Evaluation

A binary operation * is defined on the set R of real numbers by x*y= x +y + 3xy for all x, yɛR.

determine whether or not * is:

Commutative?
Associative?

The operation on the set R of real numbers is defined by a b = a+b + ab for abϵR.

Show that the operation is commutative but not associative on R.

General Evaluation

The operation * on the set R of real numbers is defined by: x * y = 3x + 2y – 1, x, yϵR.

Determine (i) 2 * 3 (ii) 1/3 * ½ (iii) -4*5

The operation * on the set R, of real numbers is defined by; p*q = p³ + q³ – 3pq; p,q ϵR. Is the operation * commutative in R?
The operation * and are defined on the set R of natural numbers by a*b = ab and a b = a/b for all a,bϵR (a) Does * distribute over ? (b) Does distribute over *?

Weekend Assignment

Two binary operation * and Ө are defined as m * n = mn – n -1 and m Ө n = mn + n -2 for al real

number m n find the value of 3 Ө (4 * 5) (a) 60 (b) 57 (c) 54 (d) 42

If x * y = x + y –xy, find x, when (x*2) + (x*3) = 63 (a) 24 (b) 22 (c) -12 (d) -21
A binary operation * is defined by a * b = a^b. If a * 2 = 2 – a, find the possible values of a (a) 1, -1

(b) 1, 2 (c) 2, -2 (d) 1, -2

The binary operation * is defined on the set of integers p and q by p*q = pq + p + q. Find 2*(3*4)

(a) 59 (b) 19 (c) 67 (d) 38

A binary operation on real numbers is defined by xy = xy + x + y for any two real numbers and y. The value of (-3/4)6 is (a) 3/4 (b) -9/2 (c) 45/4 (d) -3/4

Theory

The operation * is defined on the set R of real numbers by a* b= a+b _ 1

for all a, b €R . 5

Is the operation * commutative in R?.

The operation * is defined on the set R of real numbers by x*y = x + y + xy/2 for all x,y €R

(a) is the operation * commutative? (b) is the operation * associative over the set R?