This document defines and provides examples of unary and binary operations in mathematics. It explains that a unary operation takes one operand as input, while a binary operation takes two operands. Common unary operations include negation, reciprocal, and absolute value. Familiar binary operations are addition, subtraction, multiplication, and division. The document also discusses properties of binary operations such as commutativity, associativity, distributivity, identity elements, and inverses.

1. By Prof. Liwayway Memije-Cruz

2. • a well-defined
collection of distinct
objects.
• all the "things" that
have common
property.
• things grouped
together with a
certain property in
common.

4. • list each element (or "member") separated by a
comma, and then put some curly brackets
around the whole thing:
• curly brackets { } are sometimes called "set
brackets" or "braces".

5. • Set of even numbers: {..., -4, -2, 0, 2,
4, ...}
• Set of odd numbers: {..., -3, -1, 1, 3,
...}
• Set of prime numbers: {2, 3, 5, 7, 11,
13, 17, ...}
• Positive multiples of 3 that are less
than 10: {3, 6, 9}

6. •Sets are the fundamental
property of mathematics.
•Graph Theory, Abstract
Algebra, Real Analysis,
Complex Analysis, Linear
Algebra, Number Theory and
many more.

7. • It's a set that contains
everything.
• Everything that is
relevant to our
question.

8. Two sets are equal if they have precisely the same
members.
Example: Are A and B equal where:
• A is the set whose members are the first four positive
whole numbers
• B = {4, 2, 1, 3}
Let's check. They both contain 1. They both contain 2. And
3, And 4. And we have checked every element of both
sets, so: Yes, they are equal!
And the equals sign (=) is used to show equality, so we
write:
A = B

9. • When we define a set, if we take pieces of that
set, we can form what is called a subset.
• Example: the set {1, 2, 3, 4, 5}
• A subset of this is {1, 2, 3}. Another subset is {3,
4} or even another is {1}, etc.
• But {1, 6} is not a subset, since it has an element
(6) which is not in the parent set.
• A is a subset of B if and only if every element of
A is in B.

10. The operand is the object that is being worked on by an
operation. Operations can be mathematical ones such as
multiplication or addition, or they can be more
sophisticated functions.
In all computer languages, expressions consist of two
types of components: operands and operators. Operands
are the objects that are manipulated and operators are
the symbols that represent specific actions. For example,
in the expression
5 + x
xand 5 are operands and + is an operator. All
expressions have at least one operand.

12. • In mathematics, a unary operation is an
operation with only one operand, i.e. a single
input. An example is the function f : A → A, where
A is a set. The function f is a unary operation on
A.
• Common notations are prefix notation (e.g. +, −,
¬), postfix notation (e.g. factorial n!), functional
notation (e.g. sin x or sin(x)), and superscripts
(e.g. transpose AT). Other notations exist as well.
For example, in the case of the square root, a
horizontal bar extending the square root sign
over the argument can indicate the extent of the
argument.

13. • In common arithmetic, the unary operators are
negation, the reciprocal, and the absolute value.
Negation involves reversing the sign of a
number. For example, the negation of 4 is -4,
and the negation of -23 is 23. The reciprocal
involves dividing 1 by the number. Thus, the
reciprocal of 4 is 1/4, and the reciprocal of -23 is
-1/23. The absolute value involves reversing the
sign of a number if it is negative, and leaving the
number unchanged if it is 0 or positive. Thus, the
absolute value of 4 is 4, and the absolute value
of -23 is 23.

14. As unary operations have only one operand they are evaluated before other
operations containing them. Here is an example using negation:
3 − −2
Here, the first '−' represents the binary subtraction operation, while the second
'−' represents the unary negation of the 2 (or '−2' could be taken to mean the
integer −2). Therefore, the expression is equal to:
3 − (−2) = 5
Technically there is also a unary positive but it is not needed since we assume
a value to be positive:
(+2) = 2
Unary positive does not change the sign of a negative operation:
(+(−2)) = (−2)
In this case a unary negative is needed to change the sign:
(−(−2)) = (+2)

16. In mathematics, a binary operation on a set is a
calculation that combines two elements of the set
(called operands) to produce another element of
the set. More formally, a binary operation is an
operation of arity (the number of arguments or
operands that the function takes) of two whose two
domains and one codomain are the same set.
Examples include the familiar elementary
arithmetic operations of addition, subtraction,
multiplication and division. Other examples are
readily found in different areas of mathematics,
such as vector addition, matrix multiplication and

18.  A binary operation is said to be commutative if
a change in the order of the arguments results
in equivalence.
 Example, multiplication on the real numbers is
said to be commutative since ∀x,y∈ R,
x+y=y+x. However, there are examples where
multiplication is not commutative. For example,
if we are given two square matrices A and B,
their product AB≠BA for all matrices A and B. In
fact, AB=BA only for certain cases, hence it is
important to note what sort of set we're talking
about for binary operations.

19. • A binary operation is said to be
associative if parentheses can be
reordered and the result is equivalent.
• Example, addition is associative since
∀x,y,z∈R, (x+y)+z=x+(y+z), for
example, (1 + 2) + 3 = 1 + (2 + 3).

20. Distributivity applies when we
combined multiplication and
addition.
 Example, on the real numbers
∀x,y,z∈R, z(x+y)=zx+zy. That is
we have distributed the term z
over the sum (x + y).

21. • An element e is said to be an identity element (or
neutral element) of a binary operation if under
the operation any element combined with e
results in the same element.
• One common example can be seen in addition
on real numbers when our identity element e is
0. That is x+e=x only when e = 0. The identity
element is not always 0 though. In multiplication
on real numbers, xe=x only when our identity
element e is 1, and if A is an mxm square matrix
then Ae=A only if e is the mxm identity matrix
Imxm.

22. • For an element x, the inverse denoted x−1 when
combined with x under the binary operation
results in the identity element for that binary
operation.
• Example, for addition on real numbers, the
identity element is 0. Hence x+x−1=0 only when
our inverse is -x since ∀x∈R, x+(−x)=0.
• For multiplication on real numbers, since our
identity element is 1, then x⋅x−1=1 only when our
inverse is 1x. However, 0∈R, however 0 has no
inverse, hence we say that x−1=1/x is a
multiplicative inverse.

23. • http://mathonline.wikidot.com/unary-and-binary-set-
operations
• http://www.maths.manchester.ac.uk/~mdc/MATH10101/2
008-09/Ch8%20Groups%20Rings%20and%20Fields.pdf
• http://mathonline.wikidot.com/associativity-and-
commutativity-of-binary-operations
• https://www.techopedia.com/definition/25643/unary-
operator-c
• https://whatis.techtarget.com/definition/unary
• https://www.futurelearn.com/courses/maths-
puzzles/0/steps/14011
• https://www.mathsisfun.com/sets/sets-introduction.html