Propositional Logic|Advanced Mathematics|Class 9

Basics of Sets|Advanced Mathematics|Class 9

A Set is nothing but a well-defined collection of objects and the objects here are called the elements of the set. In real cases, some collections are as follows-

Planets of the solar system
Names of days in a week
Classes in a school

Mathematically, a set is represented by the capital letters of English alphabets, e.g., A, B, C etc. and the elements of a set is represented by the small letters of English alphabets, e.g., $a, b, c$ etc.
If $a$ and $b$ are two elements of a set A, then the set A is represented as
A = { $a, b$ }; $a, b \in{A}$
Here, $\in$ (epsilon) is a Greek letter means ‘belong to’
If $1, 2, 3$ are the elements of a set B then the set B is represented as
B = { $1, 2, 3$ }, where, $1, 2, 3 \in B$ but $4 \notin B$ i.e., the symbol $\notin$ means 'not belong to'.

Some Special Sets

N: the set of all the natural numbers: All the counting numbers, viz., { $1,2,3,4,5….$ }
W: the set of all the whole numbers: All the natural numbers including zero (no fraction/decimal), viz., { $0, 1, 2, 3….$ }
Z: the set of all the integers: All the counting numbers including zero and the negative of the same, viz.,{ $0, \pm1, \pm2, \pm3, ....$ }
Q: the set of all the rational numbers: p/q; q is not equal to zero; (fraction); viz., { $½,¾,⅗,......$ }
R: the set of all the real numbers: All the numbers that can be represented on the number line.
C: the set of all the complex numbers: The numbers containing real and imaginary parts.

Representation of a Set

A set can be represented in two forms as follows-

Tabular or Roster form

In this type of formation, all the elements of a set are listed, the elements are being separated by commas and are enclosed within the braces {}, e.g. if E be the set of all the even numbers, then E can be represented in tabular form as-
E = {

$2, 4, 6, .....$ }
Specifications:

Order of the elements doesn't matter in the roster form, e.g., let a set be S = {1, 2, 3} then the set S can also be represented as S = { $3, 2, 1$ }.
Repetition of elements is not considered in the roster form, e.g., let a set be P = {p, e, o, p, l, e} then it will be represented as P = { $p, e, o, l$ }.

Rule or Set-builder form

In this type of formation, all the elements of a set possess a single common property which is not possessed by any elements outside the set, e.g., if V be a set of vowels in English alphabet then V can be represented in Rule form as-
V = {

$x: x$ is a vowel of English alphabet}

Representation of some sets in two forms:

(i) The set of all the odd numbers up to 10.
A = {1, 3, 5, 5, 7, 9}
A = {

$x: x$ is an odd number,

$x < 10$ }

(ii) The set of even prime number.
B = {2}
B = {

$x: x$ is the even prime number}

(iii) The set of all the prime numbers up to 20.
C = {2, 3, 5, 7, 11, 13, 17, 19}
C = {

$x: x$ is a prime number,

$x < 20$ }

(iv) The set of all composite numbers up to 20.
D = {4, 6, 8, 9, 10, 12, 14, 16, 18}
D = {

$x: x$ is a composite number,

$x < 20$ }

(v) The set of natural numbers in between 5 and 15
E = {6, 7, 8, 9, 10, 11, 12, 13, 14}
E = {

$x: x$ is a natural number,

$5 < x < 15$ }

Cardinality of a set

The cardinality or cardinal number of a set is simply the size of that set. For a finite set, it is the number of elements (or members) the set contains. In symbolic notation, the size of a set 'S' is written as |S| or

$n$ (S).
For example, if S = {

$a, b, c$ } then,

$n$ (S) = 3

$\bullet$ In this case, repetition is also not considered i.e., unique elements are considered only e.g.,
if B = {

$b, o, o, k$ } then

$n$ (B) = 3
if C = {

$c, o, l, l, e, g, e$ } then

$n$ (C) = 5

Empty set

A set which contains no element is called an empty set (or void set or null set). The symbol

$\phi$ (phi) is used to represent a null set. e.g.,
A ={

$x: 5 < x < 6, x\in{N}$ }

$\bullet$ The null set is a subset of every set.

Let us check whether the following sets are empty:

A = { $x: x\in{N}$ and $(x-1) (x-2) =0$ }
B = { $x:\in{N}$ and $x^2 = 4$ }
C = { $x: x\in{N}$ and $2x-1=0$ }
D = { $x: x\in{N}$ and $x > 6$ }
E= { $x: x\in{N}$ and $x = 2n$ }

Finite set and infinite set

A set which is empty or consists of a definite number of elements is called a finite set, otherwise the set is an infinite one.
For example, let A and B are two sets and
A = {all the natural numbers} and B = {

$1 ,2, 3, 4, 5$ }
Here n(A) = infinite, so the set A is an infinite one.
And n(B) = 5, so the set B is a finite set.

Check whether the following sets are finite:

A = { $x: x\in{N}$ and $(x-1) (x-2) =0$ }
B = { $x:\in{N}$ and $x^2 = 4$ }
C = { $x: x\in{N}$ and $2x-1=0$ }
D = { $x: x\in{N}$ and $x > 6$ }
E= { $x: x\in{N}$ and $x = 2n$ }

Equal sets

Two sets A and B are said to be equal if they have exactly the same elements and we write

$A=B$ . For two sets to be equal, the order of the elements need not be the same, e.g., A = {

$1, 2, 3$ } & B ={

$3, 2, 1$ }
Here

$A=B$

Equivalent sets

Two sets A and B are said to be equivalent if n(A) = n(B), e.g., let A = {

$1,2,3,4$ } & B = {

$a, b, c, d$ }
Here, n(A) = n(B) = 4, so the sets A and B are equivalent.

Singleton set

A set is called singleton if it contains only one element, e.g., (i) A = {a} and (ii) the set containing the even prime number, i.e., {2}
Check whether the following sets are singleton:

A = { $x: x\in{N}$ and $(x-1) (x-2) =0$ }
B = { $x:\in{N}$ and $x^2 = 4$ }
C = { $x: x\in{N}$ and $2x-1=0$ }
D = { $x: x\in{N}$ and $x > 6$ }
E= { $x: x\in{N}$ and $x = 2n$ }

Subset and Superset

A set

$A$ is said to be a subset of the set

$B$ (

$A\subseteq{B}$ ) if all the elements of

$A$ are also belonging to the set

$B$ . And in such case,

$B$ is called the superset of

$A$ (

$B \supseteq{A}$ ).
Mathematically,

$A\subseteq{B}$ iff

$x\in A ⇒x\in B$

Set A is called a subset of set B if all the elements of set A are elements of set B also. If any of the elements of set A are represented by $a$ then the definition is represented by symbol $a\in{A}$ and $a\in{B}$ , symbolically, A ⊆ B (Where '⊂' means 'subset of'). The converse is also true, i.e., if A ⊆ B and $a\in{A}$ , then $a\in{B}$ . When A ⊆ B, then B ⊇ A (where ‘⊇’ means 'superset of').
On the other hand, if A is not a subset of B, it is represented by A ⊄ B.
If A ⊂ B, it is not at all understood that all the elements of B will also be the elements of the set A, however, if this happens, i.e., A ⊂ B and B ⊂ A, then it means that A = B, and it is represented by, A ⊂ B and B ⊂ A ⟺ A = B; Where, ⟺ represents if and only if (iff).

The above situation gives great insight. Since A = B, it means that it is every set is a subset of the set itself. We know that null sets or empty sets that is denoted by ϕ doesn’t contain any elements. As discussed above, a null set would be a subset in itself. Since it has no elements, it is also a subset of each and every other empty sets. This means that all the non-empty set will have at least 2 subsets: the empty set and the set itself.

If A ⊆ B and A ≠ B are, then A is called the proper subset of B (symbolically, A ⸦ B) and B ⸧ A [the proper superset], for example, N and Z represent the set of natural numbers and sets of integers respectively, we can write that N ⊂ Z. Here, N is the proper subset of Z and Z are called proper supersets of N since all the natural numbers are integers.

Venn Diagram

A Venn diagram is a way of depicting the relationship between sets. It is also called Venn-Euler diagram. In a Venn diagram a set is generally denoted by some closed curve such as circle, ellipse, triangle, or rectangle etc. For universal sets usually rectangles are chosen. John Venn was the first man who represented a set in a diagram by grouping its elements. A Venn diagram is also an easy tool in determining the validity of logical conclusions.
The diagrammatic representation for Venn diagram is shown in the above figures where each set is shown as a circle and the circles overlap if the sets intersect. So, the following are Venn diagrams for intersections and unions of two sets. The shaded parts of the diagrams are the intersection and union respectively.

Complement of a set

The complement of a set S is the collection of objects in the universal set that are not in S. The complement is written as S

$^c$ or S'.
In curly brace notation, S' = {

$x: x\in{U}, x\notin{S}$ }
S' can be found as, S' =

$U-S$
However, it should be apparent that the complement of a set always depends on which universal set is chosen.
For example, if U = {1,2,3,4,5,6,7,8,9} and A = {2,4,6,8}
then, A' =

$U-A$ = {1,3,5,7,9}

Algebraic Operations on Sets

Union of sets

The union of two sets A and B is a set consists of all the elements of A and all the elements of B, the common elements being taken only once. The symbol $\cup$ is used to denote the union.
Symbolically, A $\cup$ B = { $x: x\in{A}$ or $x \in{B}$ }.
For example, if A = {1, 2, 3}, B = {a, b, c, d}, C = {2, 3, 4} and D = {a, c, d}
Then, A $\cup$ B = {1, 2, 3, a, b, c, d}
A $\cup$ C = {1, 2, 3, 4}
A $\cup$ D = {1, 2, 3, a, c, d}
B $\cup$ D = {a, b, c, d} etc. [here, D ⊂ B]
Let P = { $x: x \in{N}$ ; $x$ is a prime number and $10 < x < 20$ } = {11, 13, 17, 19}
Q = { $x: x \in{N}; 1 < x^2 < 29$ }
= {2, 3, 4, 5}
Then P $\cup$ Q = {2, 3, 4, 5, 11, 13, 17, 19}
Note:
$\bullet$ If $x \in A\cup{B}$ , then $x\in{A}$ or $x \in{B}$ and conversely, if $x \in{A}$ or $x \in{B}$ , then $x \in{A\cup{B}}$ .
Symbolically, $x\in A\cup{B}$ $\Leftrightarrow{x \in{A}}$ or $x\in{B}$
$\bullet$ If $x \notin{A \cup{B}} \Leftrightarrow x \notin$ A and $x \notin{B}$
$\bullet$ A $\cup$ B $\cup$ C = { $x: x \in A$ or $x \in B$ or $x \in C$ }

Intersection of sets

The intersections of two sets A and B is a set, which is a collection of all the elements which are common in the sets A and B. The symbol $\cap$ is used to denote the intersection.
Symbolically, A $\cap$ B = { $x: x \in{A}$ and $x \in{B}$ }
For example, if A = {1, 2, 3}, B = {a, b, c, d}, C = {2, 3, 4} and D = {a, c, d}
Then, A $\cap$ B = {}; [intersection is a null set]
A $\cap$ C = {2}
B $\cap$ D = {a, c, d}
A $\cap$ D = {}; [intersection is a null set] etc.
Note:
$\bullet$ $x \in{(A\cap{B})} \Leftrightarrow{x \in{A}}$ and $x \in{B}$
$\bullet$ If $x \notin{(A\cap{B})} \Leftrightarrow{x \notin{A}}$ or $x \notin{B}$
$\bullet$ A $\cap$ B $\cap$ C = { $x: x \in{A}$ and $x \in{B}$ and $x \in{C}$ } [comma(,) can also be used instead of 'and'].

Disjoint sets

Two sets A and B are said to be disjoint if A

$\cap$ B =

$\phi$ . Venn diagram for two disjoint sets is shown in the following figure.
For example, Let A = {1, 2, 3,}, B = {4, 5} and C = {6, 7, 8, 9}, then
A

$\cap$ B =

$\phi$
B

$\cap$ C =

$\phi$
A

$\cap$ C =

$\phi$
Here, intersection of any two sets is a null set. Such sets A, B, C are called pairwise disjoint.

Difference of two sets

The difference between two sets A and B is a set A-B, every element of which belongs to A but does not belong to B.
Symbolically, A-B = { $x: x \in{A}, x \notin{B}$ } and B-A = { $x: x \in{B}, x \notin{A}$ }
The adjacent figure represents the difference between two sets in Venn Diagram.
From Venn diagram we see that,
A-B = A $\cap$ B'; where B' = { $x\in{U}: x\notin{B}$ }
Similarly, B-A = B $\cap$ A'; where A' = { $x\in{U}: x\notin{A}$ }

For example, if A = {1, 2, 7, 9, 0}, B = {3, 5, 7, 9}, then
A-B = {1, 2, 0}, B-A = {3, 5} and A-A = $\phi$
In general, A-B ≠ B-A
Note:
$\bullet$ $x\in{A-B} \Leftrightarrow{x\in{A}}$ and $x\notin{B}$
$\bullet$ A-B ≠ B-A (if A ≠ B)
$\bullet$ A-B ⊂ A and A-B ⊂ A $\cup$ B
$\bullet$ X-A = A’ (A ⊂ X)
$\bullet$ A = A-B $\Leftrightarrow{A\cap{B}= \phi}$
$\bullet$ A-A = $\phi$ , A- $\phi$ = A
$\bullet$ A-B, A $\cap$ B and B-A are pairwise or mutually disjoint.

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