A set is a well define collection of object.
Greek symbol ∈ (epsilon) is used to denote the phrase ‘belongs to’.
If ‘a’ is an element of a set A, we write a ∈ A.
If ‘b’ is not an element of a set A, we write b ∉ A and read “b does not belong to
A”.
There are two methods we use to represent a set
a. Roaster or Tabular form: We use commas to separate elements of sets within
braces { } without repeating elements.
For example, the set of all even positive integers less than 5 is described in roster
form as {2, 4,}.
b. Set builder form: We use this form if all the elements of a set possess a single
common property which is not possessed by any element outside the set.
Example: V = {x : x is a vowel in English alphabet}
V is read as “the set of all x such that x is a vowel of the English alphabet”. In this
description the braces stand for “the set of all”, the colon stands for “such that”.
The set of all natural numbers which divide 42 is written as A= {x : x is a natural
number which divides 42}
Some useful problems.
Q. Write the set {x: x is a positive integer and x2 < 40} in the roster form.
Solution: since x2 < 40 then, x = 1, 2, 3, 4, 5, 6.
x = {1, 2, 3, 4, 5, 6)
Q. Write the set A = {1, 4, 9, 16, 25…} in set-builder form
Ans: 1, 4, 9, 16, 25… are square of 1, 2, 3, 4, 5… respectively. Or Square of natural
number
A = {x: x is the square of a natural number}
Or,
2
A = { x : x = n , Where n ∈ N}
Q. Write the set {1/2 , 2/3, ¾, 4/5, 5/6, 6/7 } in the set-builder form
Solution We see that each member in the given set has the denominator
one more than the numerator
Also, the numerators begin from 1 and do not exceed 6.
{x : x = n/n+1
Finite and Infinite set
Let us see new set: B = { x : x is a student presently studying in both Classes X and
XI }
We observe that a student cannot study simultaneously in both Classes X and XI.
Thus, the set B contains no element at all. Such type of set is called Empty Set.
A set which does not contain any element is called the empty set or the null set or
the void set. The empty set is denoted by the symbol φ or { }
Let us see another new set
A = {men living presently in your town}
here we can know the numbers of element of this set
Let us consider the set A = {1, 2, 3, 4, 5}
This set has a definite number of elements
Such type of sets ate called finite sets respectively.
Hence, A set which is empty or consists of a definite number of elements is called
finite otherwise, the set is called infinite.
Note : All infinite sets cannot be described in the roster form. For example, the
set of real numbers cannot be described in this form, because the elements of
this set do not follow any particular pattern.
Some useful problems.
Q. State which of the following sets are finite or infinite:
(i) {x : x ∈ N and (x – 1) (x –2) = 0}
(ii) {x : x ∈ N and x² = 4}
(iii) {x : x ∈ N and 2x –1 = 0}
(iv) {x : x ∈ N and x is prime}
(v) {x : x ∈ N and x is odd}
Solution (i) Given set = {1, 2}. Hence, it is finite.
(ii) Given set = {2}. Hence, it is finite.
(iii) Given set = φ. Hence, it is finite.
(iv) The given set is the set of all prime numbers and since set of prime numbers is
infinite. Hence the given set is infinite
(v) Since there are infinite numbers of odd numbers, hence, the given set is
infinite.
Equal Sets
Two set A and B are said to be equal if they have exactly the same elements and
we
write A = B. Otherwise, the sets are said to be unequal
Some useful problems.
Q. Find the pairs of equal sets, if any, give reasons:
A = {0}, B = {x : x > 15 and x < 5},
C = {x : x – 5 = 0 }, D = {x: x² = 25},
E = {x : x is an integral positive root of the equation x² – 2x –15 = 0}.
Solution: Since 0 ∈ A and 0 does not belong to any of the sets B, C, D and E, it
follows that, A ≠ B, A ≠ C, A ≠ D, A ≠ E.
Since B = φ but none of the other sets are empty. Therefore B ≠ C, B ≠ D and B ≠ E.
Also C = {5} but –5 ∈ D, hence C ≠ D.
Since E = {5}, C = E. Further, D = {–5, 5} and E = {5}, we find that, D ≠ E.
Thus, the only pair of equal sets is C and E.
Subsets
Consider the sets : X = set of all students in your school, Y = set of all students in
your
class.
We note that every element of Y is also an element of X; we say that Y is a
subset of X. The fact that Y is subset of X is expressed in symbols as Y ⊂ X. The
symbol ⊂ stands for ‘is a subset of’ or ‘is contained in’.
Definition : A set A is said to be a subset of a set B if every element of A is also an
element of B
Q Specify subsets of the following
(a) Z=the set of integers= {0,1,-1,2,-1,3,-3,...}
(b) N=the set of nonnegative integers or natural numbers
(c) Z+=the set of positive integers
(d) Q=the set of rational numbers= {a/b| a, b is integer, b not zero}
(e) Q+=the set of positive rational numbers
(f) Q*=the set of nonzero rational numbers
(g) R=the set of real numbers
(h) R+=the set of positive real numbers
(i) R*=the set of nonzero real numbers
(j) C=the set of complex numbers
Operations on sets:
Unions
The union of A and B is denoted by A ∪ B
Two sets can be "added" together. The union of A and B, denoted by A ∪ B, is the
set of all things that are members of either A or B.
Examples:
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{1, 2} ∪ {1, 2} = {1, 2}.
{1, 2} ∪ {2, 3} = {1, 2, 3}.
Some basic properties of unions:
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A ∪ B = B ∪ A.
A ∪ (B ∪ C) = (A ∪ B) ∪ C.
A ⊆ (A ∪ B).
A ∪ A = A.
A ∪ ∅ = A.
A ⊆ B if and only if A ∪ B = B.
Intersections
A new set can also be constructed by determining which members two sets have
"in common". The intersection of A and B, denoted by A ∩ B, is the set of all
things that are members of both A and B. If A ∩ B = ∅, then A and B are said to
be disjoint.
The intersection of A and B, denoted A ∩ B.
Examples:
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{1, 2} ∩ {1, 2} = {1, 2}.
{1, 2} ∩ {2, 3} = {2}.
Some basic properties of intersections:
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A ∩ B = B ∩ A.
A ∩ (B ∩ C) = (A ∩ B) ∩ C.
A ∩ B ⊆ A.
A ∩ A = A.
A ∩ ∅ = ∅.
A ⊆ B if and only if A ∩ B = A.
Complements
Two sets can also be "subtracted". The relative complement of B in A (also called
the set-theoretic difference of A and B), denoted by A \ B (or A − B), is the set of
all elements that are members of A but not members of B.
The relative complement of B in A
The complement of A in U
The symmetric difference of A andB
All sets under discussion are considered to be subsets of a given universal
set U. In such cases, U \ A(or U-A) is called the absolute complement or
simply complement of A, and is denoted by A′.
Examples:
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{1, 2} - {1, 2} = ∅.
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{1, 2, 3, 4} - {1, 3} = {2, 4}.
If U is the set of integers, E is the set of even integers, and O is the set of
odd integers, then U - E = E′ = O.
Some basic properties of complements:
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A - B ≠ B - A for A ≠ B.
A ∪ A′ = U.
A ∩ A′ = ∅.
(A′)′ = A.
A - A = ∅.
U′ = ∅ and ∅′ = U.
A - B = A ∩ B′.
Cartesian product[
A new set can be constructed by associating every element of one set with every
element of another set. The Cartesian product of two sets A and B, denoted
by A × B is the set of all ordered pairs (a, b) such that a is a member of A and b is a
member of B.
Examples:
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{1, 2} × {red, white} = {(1, red), (1, white), (2, red), (2, white)}.
{1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2,
white), (2, green) }.
{1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
Some basic properties of cartesian products:
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A × ∅ = ∅.
A × (B ∪ C) = (A × B) ∪ (A × C).
(A ∪ B) × C = (A × C) ∪ (B × C).
Let A and B be finite sets. Then
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| A × B | = | B × A | = | A | × | B |.