The document defines binary operations as rules that assign elements of a set to ordered pairs of elements from the same set. It provides examples illustrating whether certain operations like addition and multiplication meet the criteria of being binary operations on different sets, along with explanations regarding closure, commutativity, and associativity. Additionally, it includes a series of examples and tables to analyze specific binary operations and their properties.

1. BINARY OPERATION
DEFINITIONS AND EXAMPLES

2. Definition 2.1
• A binary operation on a set S is a
rule that assigns to each ordered pair
of elements of S some element of S.

3. SOME WORDS OF WARNING
1. Exactly one element is assigned to
each possible ordered pair of
elements of S.
2. For each ordered pair of elements
of S, the element assigned to it is
again in S.

4. EXAMPLE 1
• Our usual addition + is a binary operation on the set
R. Our usual multiplication is a different binary
operation on R. In this example, we could replace R
by any of the set C, Z, R+
or Z+
.
Example 2
• Let M(R) be the set of all matrices with real entries.
The usual matrix addition + is not a binary operation
on this set.
A + B is not a defined for an ordered pair of
matrices of different shapes.
Reason:

5. Example 3
• Our Usual addition + is not a binary operation
on the set R* of nonzero real numbers.
is not in the set R*; that is, R* is not closed
under addition
Reason:
Example 4
The usual matrix multiplication is a binary operation
on the set of all 4 x 4 matrices, and the product is
again a 4 x 4 matrix.

6. Example 5
• Let F be the set of all real-valued functions f having as
domain the set R of real numbers. We are familiar with
binary operations + and on F; namely, for each ordered
pair of functions in F, we define to be the function
with domain R such that
Similarly, we define define by
Both h and k are again real-valued functions with
domain R, so that F is closed under both + and

7. Example 6
• On we define a binary operation by equals
the smaller of or or the common value if .
Thus

8. Example 7
• On define a binary operation by
Thus

9. Example 8
• On define a binary operation by where is
defined in example 6.
Thus

10. Commutative and Associative Operation
• A binary operation on a set S is
commutative if for all
• A binary operation on a set S is
associative if for all

11. Example 9
• Table 1.1 defines the binary operation on by the
following rule:
Thus and so is not commutative.
Table 1.1
a b C
a b c b
b a c b
c c b a

12. Example 10
• Complete table 1.2 so that is a commutative
binary operation on the set
Table 1.1
* a b c D
a b
b d a
c a c d
d a b b c

13. Example 11
Determine whether the definition of * does gives a
binary operation on the set.
1. On Q, define * by a*b = a/b
2. On Q+
define * by a*b = a/b
3. On Z+
,define * by a*b = a/b
4. Let F be the set of all real-valued functions with
domain R. Suppose we “define” * to give the
usual quotient of f by g, that is, f*g=h, where
h(x)=f(x)/g(x)