Education Is The Key To Success: Maths For Class XSSC Part 1 and 2

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Unit 17: SETS AND FUNCTIONS
Explanation For Exercise 17.2

OPERATIONS ON SETS

Union
Intersection
Difference
Complement

Union Of Two Sets:
The union of two sets X and Y, denoted as XUY, is the set of all those elements which belong to X or to Y or to both X and Y.
i.e., XUY = {x | x ∈ X ∨ x ∈ Y}
For example:
If X = {1,3,5} and Y = {1,2,3,4} then XUY = {1,2,3,4,5}

Intersection Of Two Sets:
The intersection of two sets X and Y, denoted as X∩Y, is the set of all those elements which belong to both X and Y.
i.e., X∩Y = {x | x ∈ X ⋀ x ∈ Y}
For example:
If X = {2,4,6,8} and Y = {l,2,3,6} then X∩Y = {2,6}

Difference Of Two Sets:
For any two sets X and Y, the difference X — Y is the set of all the elements which belong to X but do not belong to Y. It is also denoted as X\Y.
i.e., X — Y = {x | x ∈ X ⋀ x ∉ Y}
Similarly, the difference Y — X is the set of all the elements which belong to Y but do not belong to X. It is also denoted as Y\X.
i.e., Y — X = {y | y ∈ Y ⋀ y ∉ X}
For example:
If X = {4,6,8,9,10} and Y = {2,4,6,8}
then X — Y ={ 9, 10}
and Y — X = {2}

Complement Of A Set:
If set A is a subset of universal set U then the complement of A is the set of all the elements of U which are not in A. It is denoted by A' or A^c.
Thus A' = U — A
i.e., A' = {x | x ∈ U ⋀ x ∉ A}
For example:
If U ={1,2,3,...,20} and A = {1,3,5,...,19}
then A' ={2,4,6,...,20}

SYMMETRIC DIFFERENCE OF TWO SETS

The symmetric difference of two sets A and B, denoted AΔB, is the set of all the elements of A or B which are not common in both the sets.
i.e., AΔB = {x | x ∈ AUB ⋀ x ∉ A∩B}
or AΔB = (AUB) — (A∩B)
For example:
If A ={1,2,3,4,5} and B ={1,3,5,7}
then AΔB = {2,4,7}

FURTHER TYPES OF SETS:

are given below:
Disjoint Sets:
Two sets A and B are called disjoint sets if they have no element common.
i.e., Two sets A and B are disjoint sets if A∩B = ∅
For example:
The sets A ={1,3,5} and B = {2,4,6} are disjoint sets.

Overlapping Sets:
Two sets A and B are called overlapping sets if there is at least one element common in both. Moreover neither of them is subset of other.
i.e., Two sets A and B are overlapping sets if A∩B ≠ ∅ and A⊈B or B⊈A
For example:
The sets A = {1,2,3,4} and B ={2,4,6,8} are overlapping sets.

Exhaustive Sets:
If two sets A and B are subsets of universal set U then A and B are called exhaustive sets if AUB= U.
For example:
If A ={1,2,3,4}, B = {1,3,5} and U = {1,2,3,4,5}
then A and B are exhaustive sets because AUB = U

Cells:
If A and B are two non-empty subsets of universal set U then A and B are called cells if they are disjoint as well as exhaustive sets.
i.e., A and B are called cells if A and B are non-empty subsets of U and
A∩B = ∅
Also AUB = U
For example:
If A = {1,3,5,7,9}, B = {2,4,6,8,10} and U = { 1,2,3,...,10) then A and B are cells.

LAWS RELATED TO OPERATIONS ON SETS

Some important laws related to the operations on sets are as under:
Identity Laws:
For any set A

AU∅ = A
AUU = U
A∩U = A
A∩∅ = ∅

Idempotent Laws:
For any set A

AUA = A
A∩A =A

Laws of the Complement:
For any set A

AUA' = U
A∩A' = ∅
(A') = A
U' = ∅ and ∅' = U

De Morgan's Laws:
For any two sets A and B:

(AUB)' =A'∩B'
(A∩B)' = A'UB'

GO TO INDEX
Unit 17: SETS AND FUNCTIONS
Explanation For Exercise 17.1

SETS:
The concept of a set is fundamental in all branches of mathematics. OR "A set is a Collection of WELL defined and DISTINCT objects".
The sets are usually denoted by A, B, C , . . . X, Y, Z.

ELEMENTS:
A set is a well defined collection of distinct objects which are called its elements. OR Any thing belongs to a Set is called an Element (or member) of the set.
The elements are usually denoted by a, b, c, . . x, y, z.

POINTS TO REMEMBER:
Well defined means that a rule can be stated which determines either an object is a member of a set or not.
Distinct means that each object of a set is different from all other objects of the set.
If a is an element of a set A, we write a ∈ A and read "a belongs to set A".
If a is not an element of a set A, we write a ∉ A and read "a does not belong to set A".
We can not repeat a member in a set.

SOME IMPORTANT SETS OF NUMBERS:
Following notations will be used for sets of numbers:

Set of Natural Numbers: N = {1, 2, 3, . . .}
Set of Whole Numbers: W = {0, 1, 2, 3, . . .}
Set of Integers: Z = {0, ±1, ±2, ±3, . . }
Set of Positive Integers: Z+ = {0, +1, +2, +3, . . }
Set of Negative Integers: Z¯ = {0, -1, -2, -3, . . }
Set of Positive Prime Numbers: P = {2, 3, 5, 7, 11, . . . }
(Numbers come in their own table only)
Set of Positive Composite Numbers: C = {4, 6, 8, 9, 10, 12, 14 . . . }
(Opposite to prime number, come in more than one table)
Set of Odd Numbers: O = {±1, ±3, ±5, . . . .}
Set of Even Numbers: E = {0, ±2, ±4, ±6, . . . .}
Set of Rational Numbers: Q = {x|x = p/q ; p, q ∈ Z, q ≠ 0}
Set of Irrational Numbers: Q' = {x|x ≠ p/q ; p, q ∈ Z, q ≠ 0}
Set of Real Numbers: R = Q U Q'

NOTE:

The above sets can also be represented by N, W, Z, E, O, P, Q and R.
R+ and R¯ will denote the set of all positive and negative real numbers, respectively.
The set of all rational, irrational and real number can not be written in tabular form.

NOTATION:
The Notation is a system of written symbols to represent numbers, amount or elements in a set.

METHODS OR FORM OF REPRESENTING A SETS:
There are three common methods or form of representing a set. These are:
(1) Descriptive Form:
A set is described by common characteristics of its elements in any common language OR A set may also be described with the help of a statement. For example :

A= Set of natural number between 5 and 10.
A = The set of first three letters of the alphabet in English.

This is called Descriptive Form.

(2) Tabular Form:
In this form we list the elements of a set within braces (a curly bracket). For example:

If a, b, c are elements of a set A Or the set of first three letters of alphabet in English, we write:
A = {a, b, c}
Set of natural number between 5 and 10.
A = {6, 7, 8, 9}

This is called Tabular Form.

(3) Set Builder Form
A set is describe by common characteristics of its elements using symbols. For example:

A set Z where elements possess a certain property or properties.
Z = {x|x is an odd integer}
It is read as "Z is the set of all x such that x is an odd integer".
Set of natural number between 5 and 10.
N = {x | x ∈ N ∧ 5 ≤ x ≤ 10}
It is read as "N is the set of natural numbers such that x is greater than or equal to 5 and x is less than or equal to 10.

This is known as Set Builder Form.

NUMBERS OF ELEMENTS:
The number of elements in a set A is written as n(A) or |A|.
If A = {a, b, c},
then n (A) = 3 = |A|.

Symbols Use In Writing Set Builder Form

S.NO.	Symbols	Meanings
1	\|	Such that
1	∈	Belongs to
2	∉	Not belongs to
3	<	Less than
4	≮	Not less than
5	>	greater than
6	≯	Not greater than
7	≥	Greater than or equal to
8	≱	Not greater than or equal to
9	≤	less than or equal to
10	≰	Not less than or equal to
11	∧	And (logical)
12	∨	Or (logical)
13	∫	Integral
14	∓	minus or plus sign
15	∑	Sum of
16	∀	For all
17	∴	Therefore
18	∵	since

SOME IMPORTANT TYPES OF SETS:
(1) Null Set or Empty Set:
A set having no element is called the null set or the empty set. This is usually denoted by ∅ or { } .
For example:

A= Set of even numbers between 5 and 6.
B = {x|x ∈ N ∧ x < 1} = ∅.

(2) Finite Set:
A set is called finite if it consists of a specific and finite number of elements that is the process of counting the elements terminate. OR A set having limited number of elements is called a finite set.
For example:

A = {10, 12, 14, ....................... 50}
B = Set of all countries of the world.
C = {1, 2, 3, 4 }
D = {a, b, c, d, e}, etc.

Remember that the empty set is considered as a finite set.

(3) Infinite Set:
A set which is not finite is called an infinite set that is the process of counting the elements of a set does not terminate. OR A Set having unlimited number of elements is called an infinite set.
For example:

P = {10, 20, 30 ................}
Q = {x | x ∈ Q ∧ 1 ≤ x ≤ 2}
A = {1, 3, 5, . . . }
B = 1, 2, 3, . . .}
C = {. . . , -3, -2, -1, 0, 1, 2, 3, . . .}

(4) Subset:
If every element of set A is also an element of B Then A is called a subset of B.
Symbolically: we write as A ⊆ B to show that A is a subset of B.
For example:

If A = {2, 3, 5, 7}, B = {1, 2, 3, 4, 5, 6, 7} and C = {6, 7, 8, 9}
then A ⊆ B but A ⊈ C (A is not subset of C)

Note:

Every set is a subset of itself.
Null or empty set, ∅ is a subset of every set.
Every non-empty set has at least two subsets.
Number of all subsets of a set of n elements is 2ⁿ.

(5) Superset:
If set A is subset of set B then B is called superset of A.
Symbolically: we write as B ⊇ A.
For example:

If X = {a, e, i, o, u} and Y = {a, b, c, ..............., z} then Y ⊇ X

(6) Equal Sets:
Two sets A and B of same order are called equal sets if all the elements of both the sets are same. OR Two sets are said to be equal if and only if they have the same elements.
Symbolically: we write A = B
Thus A = B iff A ⊆ B and B ⊆ A
Also A = B iff A ⊇ B and B ⊇ A
For example:

Let A = {1, 2, 3, 6}, B = {x | x ∈ N Λ x ≤ 6} and C = Set of division of 6
Here A = C but A ≠ B
sets A = {a, b, c, d} and B = {b, c, a, d} are equal since they have the same elements.
We write A = B, if A and B are equal.
If C = {a, b}, then A ≠ C . (Because the do not have same elements.)
If D = {a, b, d}, then A ≠ D . (Because the do not have same elements.)

Note: Order of set A means the number of elements of set A. It is denoted by O(A) or n(A) or |A|.

(8) Equivalent Sets:
Two sets A and B are said to be equivalent sets if their orders are equal. OR
Two sets A and B are said to be equivalent if a (1-1) correspondence can be established between their elements (i.e. they have same number of elements).
Symbolically: we write as A～B, i.e., A ~ B iff O(A) = O(B)
Thus, if A ～B then one-one correspondence between their elements can be established.
In case of finite sets, this means that number of elements in any one set is the same as the number in the other Set.
For example:

Let A = {x | x ∈ R ∧ x² = 64} and B = Set of prime number less than 5.
Here A ~ B because O(A) = O(B) = 2.
Consider, A = {a, b, c, d, e}, B = {2, 3, 1, 5, 4} and C = {x, y, z, u, w}
As, n (A) = n (B) = n (C) = 5
So, A~B , B~C and C~A.
Therefore A ∼ B ∼ C.
Now consider the following sets :
P = {1, 0, 3} and Q = {3, 2, 1, 4}
Here, 1 ↔ 3, 0 ↔ 2, 3 ↔ 1 but 4 ∈ Q is left without being associated with an element of P so the sets P and Q are not equivalent and we write P ≠ Q.

Note: If two sets are equal they are also equivalent. But two equivalent sets may not be equal.

(9) Proper Subset:
Let set A is a subset of set B. The set A is called proper subset of B if A ≠ B (i.e. if there is exist at least one element in B that is not in A.)
Symbolically: we write A ⊂ B
For example:

If A = {2, 4, 6} and B = {2, 4, 6, 8}
then A ⊂ B because A is subset of B and A ≠ B.

(10) Improper Subset:
Let set A is a subset of set B. The set A is called improper subset of B iff A = B. OR
If A ⊆ B and B ⊆ A, then sets A and B are said to be improper subsets of one another. Or A set is called improper subset, if both sets are one by one equivalent to each other.
A ⊆ B and B ⊆ A implies that A = B
For example:

If set A = {a, b, c} and
set B = Set of first three letters of English alphabets.

then A is improper subset of B i.e. A = B,

Each set A is an improper subset of itself. In fact, the only improper subset of a set is the set itself.
Note: The above conditions are sometimes taken as the definition of equal sets.

(11) Power Set
The set of all the possible subsets of a set A is called Power Set of A and is denoted by P(A).
For example:

If A = {x, y, z}, then
P (A) = [ ∅, {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z}]
For set ∅, P(∅) = { ∅ }
Note: Power set of an empty set is non-empty (not empty). It consists of one element, namely, the set ∅ itself.

Consider the following:

If A = { }, P(A) = { ∅ } Here n(A) = 0 ⇒ n[P(A)] = 2° = 1
If A = (a), P(A) = { ∅, {a}} Here n(A) = 1 ⇒ n[P(A)] = 2¹ = 2
If A = (a, b), P(A) = { ∅, {a}, {b}, {a,b}} Here n(A) = 2 ⇒ n[P(A)] = 2² = 4
If A = (a, b, c} P(A) = { ∅, {a}, {b}, {c}, {a, b}, {c, a}, {b, c}, {a, b, c}} Here n(A) = 3 ⇒ n[P(A)] = 2³ = 8

From these examples we conclude that if n(A) = n elements in a set then the number of subsets of A or n[P(A)] is 2ⁿ.
The total numbers of subset is find out by 2ⁿ. Where "n" = number of elements in a set.

(12) Singleton Or Unit Set:
A set having single element is called singleton or unit set.
For example: