Download Binary Operations - I

Binary Operations
Definition:
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A binary operation on a nonempty set A is a
mapping defined on AA to A, denoted by f :
AA  A.
Binary Operation
Ex1. (a)
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Let “+” be the addition operation on Z.
+:ZZ  Z defined by +(a, b) = a+b
Let “” be the multiplication on R.
: RR  R defined by (a, b) = ab
Binary Operation
Ex1. (b)
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:ZZ  Z defined by (x, y) = x+y1
(1, 1) =
(2, 3) =
Then “” is a binary operation on Z.
∆:ZZ  Z defined by ∆(x, y) = 1+xy
∆(1, 1) =
∆(2, 3) =
Then “∆” is a binary operation on Z.
Binary Operation
Ex1. (c)
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Let “÷” be the division operation on Z.
Then ÷(1, 2)=½. (1, 2)ZZ , but ½Z.
Thus “÷” is not a binary operation.
If we deal with “÷” on R , then “÷” is not a
binary operation, either.
Because ÷(a , 0) is undefined.
But ÷ is a binary operation on R{0}.
Binary Operation
Ex2.
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The intersection and union of two sets
are both binary operations on the
universal set .
Binary Operation
Definitions:
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If “” is a binary operation on the nonempty
set A, then we say “” is commutative if
x  y = y  x, x, yA.
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If x  (y  z) = (x  y)  z,  x, y, z  A,
then we say that the binary operation is
associative.
Binary Operation
Ex3.(a)
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The Operations “+” and “” on Z are
both commutative and associative.
Binary Operation
Ex3. (b)
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But operation –:ZZZ defined by
–(a, b) = a – b is not commutative.
Since
The operation “–” is not associative, either.
Because
Binary Operation
Ex4. (a)
Let “” be the operation defined as Ex1(b)
on Z, x  y = x+y1. Then “” is both
commutative and associative.
Pf:

Binary Operation
Ex4. (b)
Let “∆” be the operation defined as Ex1(b)
on Z, x∆y = 1+xy. Then “∆” is commutative
but not associative.
Pf:
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Binary Operation
Definition:
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Let : AA  A is a binary operation
on a nonempty set A and let B  A.
If xyB, x, y B, then we say B is
closed with respect to “”.
Binary Operation
Ex5.
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(a) The set S of all odd integers is closed
with respect to multiplication.
(b) Define :ZZ  Z by x  y =x+ y.
Let B be the set of all negative integers.
Then B is not closed with respect to “”,
Binary Operation
Definition:

Let A be a nonempty set and
let : AA  A be a binary operation on A.
An element e A is called an (two side)
identity element with respect to “”
if ex = x = xe, xA.
Binary Operation
Ex6.
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(a) The integer 1 is an identity w. r. t. “”,
but not w. r. t. “+”.
The number 0 is an identity w. r. t. “+”.
(b) Let “” be the operation defined as
Ex1(b) on Z, x  y = x+y 1. Then
Binary Operation
Ex6. (continuous)
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(c) Let “∆” be the operation defined as
Ex1(b) on Z, x∆y = 1+xy. Then the
operation has no identity element in Z.
Pf:
Binary Operation
Definition:

Let e be the identity element for the binary
operation “” on A and a A.
If b A such that ab = e (or ba = e)
then b is called a right inverse
(or left inverse) of a w. r. t. .
If both a b = e = b a, then b (denoted by
a1) is called an (two-side) inverse of a;
a1 is called an invertible element of a.
Binary Operation
Note:
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The identity e and the two-side inverse of
an element w. r. t. a binary operation  are
unique.
Pf:
Binary Operation
Ex7.
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Let “” be the operation defined as Ex1(b)
on Z, x  y = x+y 1. Then (2–x) is a twoside inverse of x w. r. t. “”, xZ.
Pf:
Binary Operation
Ex8. (a)
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Give a binary operation on Z as follow.
(a) x  y = x
Binary Operation
Ex8. (b)
(b) x  y = x+2y. This operation is neither
associative, nor commutative.
Pf:
Binary Operation
Ex8. (b) (continuous)
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(b) x  y = x + 2y.This operation has no
identity, thus no inverse.
Pf:
Binary Operation
Ex8. (c)
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(c) x  y = x + xy +y.
Binary Operation