Binary Operations Properties SS3 Mathematics Lesson Note

Definition

A binary operation is any rule of the combination of any two elements of a given non-empty set. The rule of the 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 *, Θ e.t.c.

Properties:

1. 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 

1. 2 * 4 
2. 3* 1 
3. 0* 3. Is the set S closed under the operation *?

Solution

1. 2* 4, i.e, x= 2,y=4

2+4-(2×4) = 6-8 = -2.

1. 3* 1 3+1-(3×1) =4-3=1
2. 0*3 = 0 + 3-(0x3) = 3

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

1. 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.

1. 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 Ѳ on the set Z of integers is defined by; a Ѳ b = 2a +3b -1. Determine

whether or not the operation is associative in Z.

Solution

Introduce another element C

Associative condition: (ab) Oc = a Ѳ (b Өс)

(ab)c = (2a+3b-1) Ө С

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

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

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

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

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

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

Since, (a e b) O c ≠ a e (b O 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

$$

\frac{1}{3}

$$

2

is the operation closed.

1. Distributive Property: If a set is closed under two or more binary operations

(*) for all a, b and c ∈ S, such that:

a * (b e c) = (a * b) e (a * c) – Left distributive

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

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

a * b = a + b – 3, a o b = 5ab for all a, b ∈ R. Does * distribute over o.

Solution 

Let a, b ∈ R

a * (b o c) = (a * b) o (a * c)

a * (b o c) = a * (5ab)

= a + 5ab – 3.

(a * b) o (a * c) = (a + b – 3) o (a + c – 3)

= 5(a + b – 3)(a + c – 3)

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

Evaluation:

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

determine whether or not * is:

1. Commutative?
2. Associative?

2. The operation ⊕ on the set R of real numbers is defined by a ⊕ b = $$

\frac{a+b}{2}

$$ + ab for a, b ∈R,

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

General Evaluation

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

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

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

Weekend Assignment

1. Two binary operations * and ⊕ are defined as m * n = mn – n – 1 and m ⊕ n = mn + n – 2 for all real numbers m n find the value of 3 ⊕ (4 * 5) (a) 60 (b) 57 (c) 54 (d) 42
2. If x * y = x + y – x²y, find x when (x * 2) + (x * 3) = 63 (a) 24 (b) 22 (c) -12 (d) -21
3. A binary operation * is defined by a * b = ab. If a * b = 22 – a, find the possible values of a (a) 1, -1 (b) 1, 2 (c) 2, -2 (d) 1, -2
4. 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

1. A binary operation ⊕ on real numbers is defined by x ⊕ y = 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 = $$

\frac{a+b}{2}

$$ – 1

for all a, b ∈R.

Is the operation * commutative in R2?

• The operation * is defined on the set R of real numbers by x * y = x + y + $$

\frac{xx}{2}

$$ for all x, y ∈R

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

Reading Assignment: 

• Read Binary Operation, Further Mathematics Project II, page 13 – 22