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Binary operations
• A binary operation on a set A is nothing more
than a function from A × A to A.
• In other words, it takes pairs of values from A
and converts them into single values from A.
• The most familiar of binary operations are
addition and multiplication. But, we don’t write:
+: ×
or ×(5, 2) = 10. We write, x+y, and 5 × 2 instead.
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Definition
Definition
A binary operation on a nonempty set A is a mapping f form A × A
to A. That is f ⊆ A × A × A and f has the property that for each
(a, b) ∈ A × A, there is precisely one c ∈ A such that (a, b, c) ∈ f .
Notation
If f is a binary operation on A and if (a, b, c) ∈ f then we have
already seen the notation f (a, b) = c. For binary operations, it is
customary to write instead
a f b = c,
or perhaps
a ∗ b = c.
Kevin James
MTHSC 412 Section 1.4 –Binary Operations
Binary operations
• For this reason we usually write binary operations
infix that is, between their arguments, rather than
prefix, before their arguments.
• Also, the symbols chosen for binary operations
are
usually reminiscent of + or × rather than letters
like f and g as we did for single variable functions.
• For a generic binary operation we will use the
symbol .
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Ex1. (a)

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)


: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
Example
Some binary operations on Z are
1
x ∗y =x +y
2
x ∗y =x −y
3
x ∗ y = xy
4
x ∗ y = x + 2y + 3
5
x ∗ y = 1 + xy
Kevin James
MTHSC 412 Section 1.4 –Binary Operations
Ex1. (c)



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.

The intersection and union of two sets
are both binary operations on the
universal set .
Binary Operation
Operation tables
• When we are given a binary operation on a
finite set, it is common to specify it in tabular
form, sometimes called a Cayley table.
• For example, if we take to be the binary
operation on {0, 1, 2, 3} defined by
a b = min(a, b) the corresponding Cayley
table is:
0 1 2 3
0
1
2
3
0
0
0
0
0
1
1
1
0
1
2
2
0
1
2
3
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Operation tables
0
1
2
3
0
0
0
0
0
1
0
1
1
1
2
0
1
2
2
3
0
1
2
3
• To find a b in such a table we look at the
value written in the intersection of the row
marked with a (to the left of the vertical bar)
and the column marked with b (above the line.)
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Properties of binary operations
Let S be a set, and a binary operation on S.
Then:
•
is associative if, for all a, b, c S:
a (b c) = (a b) c
•
is commutative if, for all a, b S:
a b=b a
• Note that the operations × and + are both
associative and commutative.
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Example
• Consider the operation of subtraction, -, on R.
• This operation is certainly not commutative
because, for instance:
5-3 3–5
(in fact, pairs of real numbers a and b for
which a - b = b - a are rather special!)
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Example
• Nor is it associative:
(3 - 5) - 2 = -2 - 2 = -4
3 - (5 - 2) = 3 - 3 = 0
• The commutative and associative properties
are universal properties of a binary operation,
and so to show that they do not hold requires
only a single counterexample (though for
subtraction it’s not hard to find many others!)
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Commutativity and Associativity
Definition
Suppose that ∗ is a binary operation of a nonempty set A.
• ∗ is commutative if a ∗ b = b ∗ a for all a, b ∈ A.
• ∗ is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c).
Example
1 Multiplication and addition give operators on Z which are
both commutative and associative.
2
Subtraction is an operation on Z which is neither
commutative nor associative.
3
The binary operation on Z given by x ∗ y = 1 + xy is
commutative but not associative. For example
(1 ∗ 2) ∗ 3 = 3 ∗ 3 = 10 while 1 ∗ (2 ∗ 3) = 1 ∗ (7) = 8.
Kevin James
MTHSC 412 Section 1.4 –Binary Operations
Identity
• Let S be a set, and a binary operation on S.
Then:
• An element e S is a left identity for , if for
all a S:
e a=a
• An element f S is a right identity for , if for
all a S:
a f=a
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Identity
Note that on :
• 0 is both a left and right identity for addition
• 1 is both a left and right identity for
multiplication
• 0 is a right identity for subtraction, but
subtraction has no left identity.
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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:

Binary Operation
Identity Element
Definition
Let ∗ be a binary operation on a nonempty set A. An element e is
called an identity element with respect to ∗ if
e ∗x =x =x ∗e
for all x ∈ A.
Example
1 1 is an identity element for multiplication on the integers.
2
0 is an identity element for addition on the integers.
3
If ∗ is defined on Z by x ∗ y = x + y + 1 Then −1 is the
identity.
4
The operation ∗ defined on Z by x ∗ y = 1 + xy has no
identity element.
Kevin James
MTHSC 412 Section 1.4 –Binary Operations
Definition:

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.


(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
Closure
Definition
Suppose that ∗ is a binary operation on a nonempty set A and that
B ⊆ A. If it is true that a ∗ b ∈ B for all a, b ∈ B, then we say
that B is closed under ∗.
Example
Consider multiplication on Z . The set of even integers is closed
under addition.
Proof.
Suppose that a, b ∈ Z are even.
Then there are x, y ∈ Z such that a = 2x and b = 2y .
Thus a + b = 2x + 2y = 2(x + y ) which is even.
Since a and b were arbitrary even integers, it follows that the set
of even integers is closed under addition.
Kevin James
MTHSC 412 Section 1.4 –Binary Operations
Right, Left and Two-Sided Inverses
Definition
Suppose that ∗ is a binary operation on a nonempty set A and that
e is an identity element with respect to ∗. Suppose that a ∈ A.
• If there exists b ∈ A such that a ∗ b = e then b is called a
right inverse of a with respect to ∗.
• If there exists b ∈ A such that b ∗ a = e then b is called a left
inverse of a with respect to ∗.
• If b ∈ A is both a right and left inverse of a with respect to ∗
then we simply say that b is an inverse of a and we say that a
is invertible.
Kevin James
MTHSC 412 Section 1.4 –Binary Operations
Example
1 Consider the operation of addition on the integers. For any
integer a, the inverse of a with respect to addition is −a.
2
Consider the operation of multiplication on Z . The invertible
elements are 1 and -1 .
Fact
Suppose that ∗ is a binary operation on a nonempty set A. If there
is an identity element with respect to ∗ then it is unique. In the
case that there is an identity element and that ∗ is associative then
for each a ∈ A if there is an inverse of a then it is unique.
Kevin James
MTHSC 412 Section 1.4 –Binary Operations
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.


(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)


(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:


The identity e and the two-side inverse of
an element w. r. t. a binary operation  are
unique.
Pf:
Binary Operation
Ex7.


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)


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)


(b) x  y = x + 2y.This operation has no
identity, thus no inverse.
Pf:
Binary Operation
Ex8. (c)

(c) x  y = x + xy +y.
Binary Operation
VanWyk’s 103
Section 5.1 Homework Problems
1. Let ∗ be the binary operation defined on the set of natural numbers N =
{0, 1, 2, 3, . . . } given by
a ∗ b = ab .
Is ∗ commutative? Is ∗ associative?
2. Show the following binary operation ∗ on S = {0, 1} is not associative:
∗ 0 1
0 1 0
1 1 1
Hint: 03 .
3. Write the table for left projection on the set S = {a, b, c}.
4. Is the following a table for a commutative binary operation? Why or why
not?
(a)
a
a c
b b
c c
b
b
c
a
c
a
a
c
a
a c
b b
c a
b
b
b
c
c
a
c
a
(b)
5. Give an argument showing the following is a table for an associative binary
operation.
a b c
a b b b
b b b b
c b b b
6. There are 39 different binary operations on a set with 3 elements. How many
commutative binary operations are there on a set with 3 elements?
Hint: Use the symmetry of the table.
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VanWyk’s 103
Section 5.1 Homework Answers
1. ∗ is not commutative since 2 ∗ 3 = 23 = 8, while 3 ∗ 2 = 32 = 9.
∗ is not associative since (2 ∗ 3) ∗ 4 = (23 ) ∗ 4 = 8 ∗ 4 = 84 = (23 )4 = 212 ,
while 2 ∗ (3 ∗ 4) = 2 ∗ (34 ) = 2 ∗ 81 = 281 .
2. 0 ∗ (0 ∗ 0) = 0 ∗ 1 = 0, while (0 ∗ 0) ∗ 0 = 1 ∗ 0 = 1, so ∗ is not associative.
3. Since α ∗ β = α, here is the table:
∗
a
b
c
a
a
b
c
b
a
b
c
c
a
b
c
4a. No, since a ∗ c 6= c ∗ a.
4b. Yes, since the table is symmetric about the main diagonal.
5. Since α ∗ β = b, it follows that for all α, β, γ ∈ S, (α ∗ β) ∗ γ = b = α ∗ (β ∗
γ). So ∗ is associative.
6. The answer is 36 . By the symmetry we need for commutativity, the table
must look like
a b c
a x u v
b u y w
c v w z
So there are really only 6 slots we can fill. (Once the 6 on and above the
main diagonal are filled in, the 3 below that are determined.)
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