1. Closure

A binary operation combines 2 two elements from one section to give a 3^rd element. If the third element is found in the original set, then the operation is closed.

Example;

1. The set N (natural numbers) is closed under “+” since the sum of any two numbers is always in N.
2. The set N is not closed under subtraction
3. The set Z (integers) is closed under addition, subtraction and multiplication

2. Commutative law

If the element of * a, b ∈ S and if there exist any elements of a*b = b*a, then the operation is commutative. If x, y ∈ S such that x*y ≠ y*x, then the operation is not commutative.

Example;

1. Addition in N is commutative since 3+2 = 2+3 → 5=5
2. Subtraction and division in N is not commutative

e.g. the operation * is defined on Z by a*b = a²b – b²a, evaluate the following;

i) a*b = a²b – b²a

   3*5 = (3)²(5) – (5)² (3)

          = 45 – 75 = -30

Ii) 3*5 = 5*3

   -30 = (5)² (3) - (3)²(5)

   -30 ≠30, then its not commutative

3. . Associative law

A binary operation is associative defined * and a set B is associative if for every element a, b ∈ R, (a*b)*c = a*(b*c).

Example;

Addition and multiplication is associative in N.

The binary operation * is denoted on the set of N such that * is x*y = 2x + y. verify if * is associative with 1*2*3.

Solution;

(1*2) *3 = 1* (2*3)          

X= 1                x = 2

Y = 2                y = 3

2(1) +2 *3 = 1* 2(2) +3

4*3 = 1*7

X= 4                x = 1

Y = 3                y = 7

2(4) + 3 = 2(1) + 7

11 = 9 hence it is not associative

4. Identity element

Let * be the binary operation of a set S, e is called identity element if and only if  a ∈ S. that is;

a*e = e*a = a

0. is the identity element for addition in Z because;

0 + 2 = 2 + 0 = 2

0 + 4 = 4 + 0 = 4

• if the identity element exists, it is unique

1. is the identity element for multiplication;
2. x 2 = 2 x 1 = 2

 Example;

A binary operation * is defined on R by a*b = a + b + 3. Find e.

Solution;

 (a*b) * e = (a*b)

(a + b +3) * e = a + b + 3

(a = a + b + 3), b= e

Then, a + b + 3→ (a + b + 3) + (e) + 3 = a + b + 3

→e = 3 – 6 = -3

5. the inverse element

the inverse of an element a under an operation * is usually denoted as a^-1 such that;

a * a^-1 = a^-1 * a = e.