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24-02-2016 12:01:00 +0000
Good day all,
The concept of Binary Operation is the most fundamental in mathematics often tested on JAMB and other exams. so lets take a quick look into this topics.
Binary Operation - Any rule of combination of any two elements of a given non -empty set.
Closure Property of Binary Operation - A non-empty set P is said to be closed under a binary operation * if for all a, b∈P, a*b∈P
Commutative Property - Given a non- empty set P which is closed under a binary operation *, if for all a,b∈P
a * b = b * a, then the binary operation * is said to be commutative.
Associative Property - If a, b, C∈P and P is closed under a binary operation *, then (a * b)* c = a * (b * c)
Distributive Property - Given that a non-empty set P is closed under the operation * and ⊕; if for all a, b, c∈P
then a * (b⊕c) = (a * b)⊕(a * c) (Left Distributive over ⊕)
(b⊕c)* a = (b * a)⊕(c * a) (Right Distributive over ⊕)
Identity or Neutral Element - Given a non- empty set which is closed under a binary operation *, if there exists an element e∈P such that x * e = e * x = x,then e is called Identity element.
Inverse Property - Given a non empty set P which is closed under a binary operation*, if for x∈P, there exist x'∈P such that x * x' = x' * x = e
where e is the Identity element in P under the operation *, then x' is called the inverse of x in P
Please feel free to ask questions where you do not understand.
Now lets try solving the questions below;
1.The binary operation (*) is defined by x * y = xy - y - x for all real values of x and y.
If (x * 3) = (2 * x), find x. (Jamb question number (22), 1998).
2. If a * b = √ab, evaluate 2 * (12 * 27). (Jamb question number (22), 1995).
3 .A binary operation * is defined by a * b = ab + a + b for any real number a and b. If the identity element is zero. Find the inverse of 2 under this operation. (Jamb question number (19), 1999).
4. If x * y = x + y - xy, find x, when (x * 2) + (x * 3) = 63. (Jamb question number (22), 1997).