BINARY OPERATIONS

BINARY OPERATIONS

• Binary – “ two quantities”
• Any operation that combines two values to create a new one.
• Common binary operations are: addition, subtraction, multiplication and division.

BINARY OPERATIONS

• Binary operation on a set is a rule for combining two elements of the set.
• A binary operation * on a set A is a function from A X A into A.
• For each (a, b) є A X A, we denote the element * (a,b) by (a *b).

Example:

• Suppose we are talking about REAL NUMBERS (a, b)
• By addition: (a, b) when added it gives you another REAL NUMBERS (5, 4), so when added 5 + 4 = 9 , nine is a real number.
• Thus, ADDITION is a binary operation when it comes to Real Numbers.

Example:

• By division: (a, b) є N if a=8 and b=2, what is a/b? a/b = 8/2 = 4 4 є N Therefore, DIVISION is a binary operation for natural numbers.

PROPERTIES OF BINARY OPERATIONS

• Commutative Property- Let S be a non-empty set. A binary operation * on S is said to be commutative, if a*b= b*a, for all a, b є N

Example:
Basic commutative binary operations are + and x

2+3= 3+ 2
5=5

5 x 2= 2 x 5
10=10

For every real number a and b, is the binary operation % is defined

a % b= 5a + 5b+(ab)^2
a % b = b % a
a % b = 5(a) + 5( b)+ (ab)^2
a % b = 5a + 5b+ a^2 b^2
b % a = 5(b) + 5( a)+ (ab)^2

b % a = 5b + 5a + b^2 a^2
b % a = 5a + 5b + a^2 b^2
a % b = b % a

Therefore, % is commutative.

PROPERTIES OF BINARY OPERATIONS

• Associative Property- Let S be a subset of Z. A binary operation * on S is said to be associative, if (a*b)*c=a*(b*c), for all a, b, c є S

Example: Given that a * b = 2a + 2b.

Evaluate i) ( 3 * 4) * 5 ii) 3 * ( 4* 5)

i) ( 3 * 4) * 5 = [ 2 (3) + 2 (4)] * 5
= [6 + 8] * 5
= 14 * 5
= 2( 14) + 2 (5)
= 28 + 10
( 3 * 4) * 5 = 38

Continuation:

ii) 3 * ( 4* 5)
= 3 * [2(4) + 2( 5)]
= 3* [ 8+ 10]
= 3 * 18
= 2 (3) + 2 (18)
= 6 + 36
3 * ( 4* 5) = 42

Since ( 3 * 4) * 5 ≠ 3 * ( 4* 5), then the binary operation * is not associative.

PROPERTIES OF BINARY OPERATIONS

• Closure Property- Let S be a non-empty set. A binary operation * on S is said to be closure binary operation on S, if a*b є S, for all a, b є S

Example:

Let U= { 1, 2, 3, 4, 5, 6}. Is the operation * defined by a * b = 2a +2b closed in U?
1 * 2 = 2(1) + 2 ( 2)
= 2 + 4
1 * 2 = 6
6 є U

3* 4 = 2(3) + 2 ( 4)
= 6 + 8
3 * 4 = 14
Therefore, U is not closed under the binary operation * since 14 is not an element of U.

PROPERTIES OF BINARY OPERATIONS

• Identity Property- A non-empty set S with binary operation *, is said to have an identity e є S, if e*a=a*e=a, for all a є S.

Example:


0 + a =a + 0 = a
0 + 5= 5 + 0 = 5
Identity element under addition of Real number is 0.

1 x a= a x 1 = a
1 x 6= 6 x 1 = 6
Identity element under multiplication of Real number is 1.

QUIZ. Answer the following questions.


A. Write Yes if it is a binary operation and No if is not.
______1. on N, (a * b) = ab
______2. on R, (a *b) = a+b
________3. on Z+ , (a *b) = a- b
_______4. On Z , ( a* b) = ab
_________5. On Z+ , ( a * b) =a/b


B. (5 pts.each)

1. Evaluate whether the binary operation (a * b) = 2a + ab is a commutative. If a= 5 and b= 3.

Thank You