Given a non- empty set S which is closed under a binary operation * and if there exists an element e € S such that a*e = e*a = a for all a € S, then e is called the IDENTITY or NEUTRAL element. The element is unique.
Example: The operation * on the set R of real numbers is defined by a*b = 2a-1┼ b
for all a, b € R. Determine the identity element.
a*e= e*a = a
a*b= 2a-1┼ b
a*e = 2a-1┼ e = a
2a-1+ 2e = 2a
2e = 2a-2a +1
e = ½.
Find the identity element of the binary operation a*b = a +b+ab
If x € S and an element x-1 € S such that x*x-1 = x-1*x= e where e is the identity element and x-1 is the inverse element.
Example: An operation * is defined on the set of real numbers by x*y = x + y -2xy. If the identity element is 0, find the inverse of the element.
X *y = x+ y- 2xy
x*x-1 = x-1*x= e, e = 0
x + x-1– 2xx-1 = 0
x-1 -2xx-1= -x
x-1(1-2x) = -x
x-1 = -x/ (1-2x)
The inverse element x-1 = -x/ (1-2x)
The operation ∆ on the set Q of rational numbers is defined by: x∆ y = 9xy for x,y € Q
Find under the operation ∆ (I) the identity element (II) the inverse of the element a € Q
1. An operation on the set of integers defined by a*b = a2 + b2 – 2a,find 2*3*4
2. Solve the pair of equations simultaneously
(a) 2x + y = 3, 4x2 – y2 + 2x + 3y= 16
(b) 22x – 3y = 4, 33x + 5y – 18 = 0
Reading Assignment: Read Binary Operation, Further Mathematics Project II, page 16 – 22
- Find the identity element e under this operation if the binary operation* is defined by c * d = 2cd+ 4c+ 3d for any real number.
A. -3 B. -2C+3 C. X-3
2. An operation is defined by x*y = Logyx , evaluate 10* 0.0001
A. 4 B. -4 C. 3
3. The binary operation * is defined by x*y= xy– 2x -15, solve for x if x*2= 0
A.x= -3 or -5 B. x= -3 or 5 C. x = 3 or 5
4. A binary operation * is defined on the set R of real numbers by
m*n = m + n2 for all m, n € R. If k*3 = 7*4, find the value of k
A. 8 B.28/3 C.14
5 .Find the inverse function a-1 in the binary operation ∆ such that for all a,b € R
a ∆ b = ab/ 5
A. 25/a B.-25/a C. a/5
- A binary operation * is defined on the set R of real numbers by
x*y = x2 + y2+ xy for all x, y € R. Calculate (a) ( 2*3)* 4
(b) Solve the equation 6*x = 27
2. Draw a multiplication table for modulo 4.
(b) Using your table or otherwise evaluate (2X3) X (3X2)Pass WAEC, JAMB, POST-UTME & more in One Sitting for FREE!💃