CONTENT
- Concept of binary operations,
- Closure property
- Commutative property
- Associative property and
- Distributive property.
Definition:
Binary operation is any rule of combination of any two elements of a given non empty set. The rule of 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, *, Ө e.t.c.
Properties:
A. 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 (a) 2 * 4 (b) 3* 1 (c) 0* 3. Is the set S closed under the operation *?
Solution
- 2 * 4, i.e, x= 2,y=4
2+ 4 – (2×4) = 6-8 = -2.
- 3* 1 = 3+1-( 3x 1) = 4 – 3= 1
- 0*3 = 0 + 3 –( 0 x3) = 3
Since -2€ S, therefore the operation * is not closed in S.
B. 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.
C. 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 Ө on the set Z of integers is defined by; a Ө b = 2a +3b -1. Determine whether or not the operation is associative in Z.
Solution
Introduce another element C
Associative condition: (aӨb) Өc = a Ө (b Өc)
(aӨb)Өc = (2a+ 3b- 1) Ө C
= 2(2a +3b -1) + 3c -1
= 4a + 6b- 2+ 3c- 1
= 4a +6b+3c- 3.
Also, the RHS, a Ө (b Ө c) = a Ө (2b+3c- 1)
= 2a+ 3(2b +3c- 1) – 1
= 2a + 6b +9c -3 -1
a Ө (b Ө c) = 2a+ 6b+ 9c -4
Since, (a Ө b) Ө c ≠ a Ө (b Ө 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
3 2
is the operation closed.
D. Distributive Property: If a set is closed under two or more binary operations
(* Ө) for all a, b and c € S, such that:
a*(bӨ c) = (a*b )Ө( a*c – Left distributive
(BӨc) *a = (b*a) Ө(c*a) – Right distributive over the operation Ө
Example: Given the set R of real numbers under the operations * and Ө defined by:
a*b = a+ b- 3, aӨb= 5ab for all a, b € R. Does * distribute over Ө.
Solution Let a, b,c € R
a* ( bӨc) = (a*b) Ө (a*c)
a* (bӨc) = a* (5ab)
= a+ 5ab -3.
(a*b) Ө (a*c) = (a+ b -3) Ө ( a+ c-3)
= 5(a +b-3)(a +c -3)
From the expansion, it’s obvious that, a* ( bӨc) ≠ (a*b) Ө (a*c) therefore * does not distribute over Ө.
Evaluation:
1. A binary operation * is defined on the set R of real numbers by x*y= x +y + 3xy for all x, yɛR.
determine whether or not * is:
- Commutative?
- Associative?
General Evaluation
1. The operation * on the set R of real numbers is defined by: x * y = 3x + 2y – 1, x, yϵR.
Determine (i) 2 * 3 (ii) 1/3 * ½ (iii) -4*5
2. The operation * on the set R, of real numbers is defined by; p*q = p3 + q3 – 3pq; p,q ϵR. Is the operation * commutative in R?
Weekend Assignment
- Two binary operation * and Ө are defined as m * n = mn – n -1 and m Ө n = mn + n -2 for al real
number m n find the value of 3 Ө (4 * 5) (a) 60 (b) 57 (c) 54 (d) 42
- If x * y = x + y –xy, find x, when (x*2) + (x*3) = 63 (a) 24 (b) 22 (c) -12 (d) -21
- A binary operation * is defined by a * b = ab. If a * 2 = 2 – a, find the possible values of a (a) 1, -1
(b) 1, 2 (c) 2, -2 (d) 1, -2
- 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
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