This document defines and provides examples of binary operations, properties of operations like commutativity and associativity, and special elements related to operations like identity and inverse elements. A binary operation on a set A is a rule that assigns exactly one element a*b in A to each ordered pair (a,b) of elements in A. Examples of properties discussed are an operation being commutative if a*b = b*a for all a,b in A, and associative if (a*b)*c = a*(b*c) for all a,b,c in A. Identity elements satisfy e*a = a*e = a for all a, and inverse elements satisfy a*x = e = x

1. CHAPTER 2: GROUPS AND SUBGROUP
2.1 Binary Operations
A binary operation (or just operation) ∗ on a set A
is a rule which assigns to each ordered pair (a,b) of elements
of A exactly one element a ∗ b in A.
That is :
∗: A X A → A
Example
1. The usual addition (+) on Z, R, C, R+, Z+.
2. The usual multiplication (*) on Z, R, C, R+, Z+.
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2. Three aspects of the definition that need to be stressed:
(1) a ∗ b is defined for every ordered pair (a,b) of
elements of A.
Addition (+) on M(R) is not defined. WHY?
M(R) -- the set of all matrices with real entries.
(2) a ∗ b must be uniquely defined.
Suppose we define an operation ⊗ R such that
on
for any a, b ∈ R, a ⊗ b is the number whose square is ab.
⊗ is not uniquely defined. WHY?
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3. (3) If a and b are in A, then a ∗ b must also be in A.
-closed under operation.
Suppose we have a set A={0,1,2,3,4}
Is + on A an operation?
Example
1. Is addition (+) an operation on R*? R*-Nonzero real numbers.
Solution
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4. Properties of Operation
Commutative
An operation ∗ on a set A is commutative if (and only if)
for all a, b ∈ A.
a∗b=b∗a
Example
Is the operation below commutative?
+
1. ∗ be an operation on Z such that for a, b ∈ Z ,
a∗b equals the smaller of a and b or the common value if a=b.
+
Solution
Remark
If the question is: Is ∗ a commutative operation ….?
Need to check whether it is an operation first!!
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5. 2.
be an operation on Z+ such that
a b =a
Solution
Associative Operation
An operation ∗ on a set A is associative if (and only if)
(a ∗ b) ∗C =
a ∗ (b ∗C )
Example
Is Addition on R associative? Is division R associative.
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6. Identitiy Element
Let ∗ be an operation on a set A.
If there is an element e in A with the property that
e ∗ a = a and a ∗ e =a
for every element a in A
then e is called an identity or “neutral” element with
respect to the operation ∗
Example
What is the identity element for addition in R?
What is the identity element for multiplication in R?
Remark
• An identity element is unique.
That is, it is the same for all element of a set.
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7. Inverse Element
Let ∗ be an operation on a set A.
If a is an element in A, and x is an element of A with the
property that
a ∗ x = e and x ∗ a =e
then x is called an inverse of a.
Example
What is the inverse of a for addition in R?
What is the inverse of a for multiplication in R,
( a ≠ 0)?
Remark
• An inverse element is not unique in a set but it is unique
for each element.
• The inverse of a is denoted by a-1.
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