Binary Operations: Basic Concept Of Binary Operations

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:

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

• 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.

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= 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.

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                                                                                                                                                                    

                       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

1. 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.

D.   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:

1.  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?

General Evaluation

1. 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

2. 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?

Weekend Assignment

1. 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