Set theory- Introduction, symbols with its meaning

Sanjivani College of Engineering, Kopargaon
Department of Electronics & Computer Engineering
(An Autonomous Institute)
Affiliated to Savitribai Phule Pune University
Accredited ‘A’ Grade by NAAC
________________________________________________________________________________________
Subject: Discrete Mathematics and Information Theory (EC 201)
UNIT-1
Topic: Set Theory
Prof. Dipak P. Mahurkar
Assistant Professor, ECE Department

Symbol Meaning and Description Example
∨ Logical OR: Represents disjunction (at least one is true). A ∨ B
∧ Logical AND: Represents conjunction (both are true). A ∧ B
¬ Logical NOT: Represents negation (opposite of the value). ¬A
→ Logical Implication: If...then... statement. If A, then B
↔ Logical Equivalence: If and only if, two statements are equal. A ↔ B
∀ Universal Quantification: "For all" or "For every". ∀x (x > 0)
∃ Existential Quantification: "There exists". ∃x (x < 0)
∈ Element of: Indicates membership in a set. x ∈ A
∉ Not an Element of: Indicates non-membership in a set. x ∉ A

Symbol Meaning and Description Example
∅ Empty Set: A set with no elements. ∅
∩ Set Intersection: Elements common to both sets. A ∩ B
∪ Set Union: All unique elements from both sets. A ∪ B
⊆ Subset: One set is entirely contained in another. A ⊆ B
⊂ Proper Subset: A subset that is not equal to the whole set. A ⊂ B
⊇ Superset: One set contains another entirely. A ⊇ B
⊃ Proper Superset: A superset that is not equal to the whole set. A ⊃ B
∞ Infinity: Represents an unbounded value. lim(x → ∞) f(x)
≡ Congruence: Two quantities are equivalent in a specific context. a ≡ b (mod n)

Set
A set is a group of “objects”
People in a class: { Alice, Bob, Chris }
Classes offered by a department: { ECE223, ECE024, ECE203 … }
Colors of a rainbow: { violet, indigo, blue, green, yellow,
orange, red}
States of matter { solid, liquid, gas, plasma }
Sets can contain non-related elements: { 3, a, red, Nagpur }
Although a set can contain (almost) anything, we will most often
use sets of numbers
4
All positive numbers less than or equal to 5: {1, 2, 3, 4, 5}
A few selected real numbers: { 2.1, π, 0, -6.32, e }

Set properties 1
5
Order does not matter
We often write them in order because it is easier for
humans to understand it that way
{1, 2, 3, 4, 5} is equivalent to {3, 5, 2, 4, 1}
Sets are notated with
curly brackets

Set properties 2
6
Sets do not have duplicate elements
Consider the set of vowels in the alphabet.
It makes no sense to list them as {a, a, a, e, i, o, o, o, o, o, u} What we really
want is just {a, e, i, o, u}
Consider the list of students in this class
Again, it does not make sense to list somebody twice
Note that a list is like a set, but order does matter in a set
We won’t be studying lists much in this class

Specifying a set 1
7
Sets are usually represented by a
capital letter (A, B, S, etc.)
Elements are usually represented by an italic
lower-case letter (a, x, y, etc.)
Easiest way to specify a set is to list all
the elements: A = {1, 2, 3, 4, 5}
Not always feasible for large or
infinite sets

Specifying a set 2
8
Can use an ellipsis (…): B = {0, 1, 2, 3, …}
Can cause confusion. Consider the set C = {3, 5, 7, …}. What comes
next?
If the set is all odd integers greater than 2, it is 9
If the set is all prime numbers greater than 2, it is 11
Can use set-builder notation
D = {x | x is prime and x > 2}
E = {x | x is odd and x > 2}
The vertical bar means “such that”
Thus, set D is read (in English) as: “all elements x such that x is prime
and x is greater than 2”

Specifying a set 3
9
A set is said to “contain” the various “members” or “elements” that make up the set
If an element a is a member of (or an element of) a set S, we use then notation
a  S
4  {1, 2, 3, 4}
If an element is not a member of (or an element of) a set S, we use the notation
a  S
7  {1, 2, 3, 4}
Virginia  {1, 2, 3, 4}

Set equality
10
Two sets are equal if they have the same elements
{1, 2, 3, 4, 5} = {5, 4, 3, 2, 1}
Remember that order does not matter!
{1, 2, 3, 2, 4, 3, 2, 1} = {4, 3, 2, 1}
Remember that duplicate elements do not matter!
Two sets are not equal if they do not have the same elements
{1, 2, 3, 4, 5} ≠ {1, 2, 3, 4}

Subsets 1
If all the elements of a set S are also elements of a
set T, then S is a subset of T
For example, if S = {2, 4, 6} and T = {1, 2, 3, 4, 5, 6, 7},
then S is a subset of T
This is specified by S  T
Or by {2, 4, 6}  {1, 2, 3, 4, 5, 6, 7}
If S is not a subset of T, it is written as such:
S  T
For example, {1, 2, 8}  {1, 2, 3, 4, 5, 6, 7}
11

Subsets 2
12
Note that any set is a subset of itself!
Given set S = {2, 4, 6}, since all the elements of S are
elements of S, S is a subset of itself
This is kind of like saying 5 is
less than or equal to 5
Thus, for any set S, S  S

Subsets 3
13
The empty set is a subset of all sets (including itself!)
Recall that all sets are subsets of themselves
All sets are subsets of the universal set
Another way to define a subset:
x ( xA  xB )
In words: For all possible values of x, (meaning for all possible
elements of a set), if x is an element of A, then x is an element of B

If S is a subset of T, and S is not equal to T, then S is a proper subset of T
Let T = {0, 1, 2, 3, 4, 5}
If S = {1, 2, 3}, S is not equal to T, and S is a
subset of T A proper subset is written as S  T
Let R = {0, 1, 2, 3, 4, 5}. R is equal to T, and thus is a subset (but not a
proper subset) or T
Can be written as: R  T and R  T (or just R = T)
Let Q = {4, 5, 6}. Q is neither a subset or T nor a proper subset of T
14
Proper Subsets 1

Proper Subsets 2
15
The difference between “subset” and “proper subset” is like the difference
between “less than or equal to” and “less than” for numbers
The empty set is a proper subset of all sets other than the
empty set (as it is equal to the empty set)

The universal set 1
16
U is the universal set – the set of all of elements (or the “universe”) from which given any set
is drawn
For the set {-2, 0.4, 2}, U would be the real numbers
For the set {0, 1, 2}, U could be the natural numbers (zero and up), the integers, the
rational numbers, or the real numbers, depending on the context

The universal set 2
17
For the set of the students in this class, U would be all the students in the
University (or perhaps all the people in the world)
For the set of the vowels of the alphabet, U
would be all the letters of the alphabet
To differentiate U from U (which is a set operation), the universal set is
written in a different font (and in bold and italics)

Complement of a Set
The complement of set A is denoted by A’ or by AC .
A’ = {x| x is not in set A}.
The complement set operation is analogous to the negation operation
in logic.
Eg:- Say U={1,2,3,4,5}, A={1,2}, then A’ = {3,4,5}.

Union of sets
The union of two sets A, B is denoted by
A U B.
A U B = {x| x is in A or x is in B}
Note the usage of or. This is similar to disjunction
A v B.
A={1,2,3,4}
B={4,5,6,7}
A U B={1,2,3,4,5,6,7}

Difference of Sets
If set A and set B are two sets, then set A difference
set B is a set which has elements of A but no
elements of B.
It is denoted as A – B.
Example: A = {1,2,3} and
B = {2,3,4}
A – B = {1}

Intersection of sets
When an element of a set belongs to two or more sets
we say the sets will intersect.
The intersection of a set A and a set B is denoted by A
∩ B.
A ∩ B = {x| x is in A and x is in B}
Note the usage of and. This is similar to conjunction. A
^ B.
Example A={1, 3, 5, 7, 9} and B={1, 2, 3, 4, 5}
Then A ∩ B = {1, 3, 5}. Note that 1, 3, 5 are in both A
and B.

A + B ={x|x  A-B OR x  B-A}
A + B = (A-B) U (B-A)
A={4,5,6,7,8,9}
B={2,3,5,7} 
A + B ={2,3,4,6,8,9}
A-B= {4,6,8,9}
B-A={2,3}

Mutually Exclusive Sets
We say two sets A and B are mutually
exclusive if A ∩ B = Φ .
Think of this as two events that can not
happen at the same time.

Set theory- Introduction, symbols with its meaning

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