Set Theory
A Set is defined as a group of objects, known as elements. These objects could be anything conceivable, including numbers, letters, colors, even set themselves. However, none of the objects of the set can be the set itself.
Set Notation
We write sets using braces and denote them with capital letters. The most natural way to describe sets is by listing all its members.
For example,
A = {1,2,3,…,10} is the set of the first 10 counting numbers, or naturals, B = {Red, Blue, Green} is the set of primary colors, N = {1,2,3,…} is the set of all naturals, and Z = {...,−3,−2,−1,0,1,2,3,…} is the set of all integers.
Well defined Set
Well-defined means, it must be absolutely clear that which object belongs to the set and which does not.
Some common examples of well defined sets:
- The collection of vowels in English alphabets. This set contains five elements, namely, a, e, i, o, u
- N = {1,2,3,…} is the set of counting numbers, or naturals.
- N = {1,2,3,…} is the set of counting numbers, or naturals.
- Z = {…,−3,−2,−1,0,1,2,3,…} is the set of integers.
Set Equality
Two sets A and B are said to be equal if and only if both the sets have same and exact number of elements. Here, if and only if means that both parts of the statement ("A = B" and "both sets have the exact same elements") are interchangeable. For example,
{2,4,6,8} = {4,8,6,2} and {2,4,6,8} = {2,4,2,6,8,2,6,4,4}.
Another example comes from the set of even naturals, which can be described as E = {2,4,6,8,…} = {2x | x ∊ N}.
Null Set
A very important set is the empty set, or the null set, which has no elements. We denote the empty set by ∅, or {}. Note that we could also write, for example, ∅= {x | x ∊N and x < 0} or ∅
= {x | x ∊Q and x ∉Q}.
Intersection of Sets
The intersection of sets A and B, denoted as A ∩ B, is the set of elements common to both A AND B.
For example:
A = {1,2,3,4,5}
B = {2,4,6,8,10}
The intersection of A and B (i.e. A∩B) is simply {2, 4}
Union of Sets
The union of sets A and B, written as A∪B, is the set of elements that appear in either A OR B.
For example:
A = {1,2,3,4,5}
B = {2,4,6,8,10}
The union of A and B (i.e. A∪B) is {1, 2, 3, 4, 5, 6, 8, 10}
Difference of Sets
The difference of sets A and B, written as A-B, is the set of elements belonging to set A and NOT to set B.
For example:
A = {1,2,3,4,5}
B = {2,3,5}
The difference of A and B (i.e. A-B) is {1,4}
NOTE: A-B ≠ B-A
Cartesian Product of Sets
The Cartesian product of sets A and B, written A x B, is expressed as:
A x B = {(a,b)│a is every element in A, b is every element in B}
For example:
A = {1,2}
B = {4,5,6}
The Cartesian product of A and B (i.e. A x B) is {(1,4), (1,5), (1,6), (2,4), (2,5), (2,6)}
Now, lets us try doing some questions based on Set Theory.
Solved Questions:
Question 1: If ∪ = {1, 3, 5, 7, 9, 11, 13}, then which of the following are subsets of U.
B = {2, 4}
A = {0}
C = {1, 9, 5, 13}
D = {5, 11, 1}
E = {13, 7, 9, 11, 5, 3, 1}
F = {2, 3, 4, 5}
Answer: Here, we can see that C, D and E have the terms which are there in ∪. Therefore, C, D and E are the subsets of ∪.
Question 2: Let A and B be two finite sets such that n(A) = 20, n(B) = 28 and n(A ∪ B) = 36, find n(A ∩ B).
Solution: Using the formula n(A ∪ B) = n(A) + n(B) - n(A ∩ B).
then n(A ∩B) = n(A) + n(B) - n(A ∪B)
= 20 + 28 - 36
= 48 - 36
= 12
Question 3: In a group of 60 people, 27 like cold drinks and 42 like hot drinks and each person likes at least one of the two drinks. How many like both coffee and tea?
Solution: Let A = Set of people who like cold drinks B = Set of people who like hot drinks Given,
(A ∪B) = 60 n(A) = 27 n(B) = 42 then;
n(A ∩ B) = n(A) + n(B) - n(A ∪ B)
= 27 + 42 - 60
= 69 - 60 = 9
= 9
Therefore, 9 people like both tea and coffee.
Question 4: In a competition, a school awarded medals in different categories. 36 medals in dance, 12 medals in dramatics and 18 medals in music. If these medals went to a total of 45 persons and only 4 persons got medals in all the three categories, how many received medals in exactly two of these categories?
Solution: Let A = set of persons who got medals in dance.
B = set of persons who got medals in dramatics.
C = set of persons who got medals in music.
Given,
n(A) = 36
n(B) = 12
n(C) = 18
n(A ∪ B ∪C) = 45
n(A ∩ B ∩ C) = 4
We know that number of elements belonging to exactly two of the three sets A, B, C
= n(A ∩ B) + n(B ∩ C) + n(A ∩ C) - 3n(A ∩ B ∩ C)
= n(A ∩ B) + n(B ∩ C) + n(A ∩ C) - 3 × 4 ……..(i)
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩C)
Therefore, n(A ∩ B) + n(B ∩ C) + n(A ∩ C) = n(A) + n(B) + n(C) + n(A ∩ B ∩ C) - n(A ∪ B ∪ C)
From (i) required number
= n(A) + n(B) + n(C) + n(A ∩ B ∩ C) - n(A ∪ B ∪ C) - 12
= 36 + 12 + 18 + 4 - 45 - 12
= 70 - 67
= 3
Question 5: In a group of 100 persons, 72 people can speak English and 43 can speak French. How many can speak English only? How many can speak French only and how many can speak both English and French?
Solution: Let A be the set of people who speak English.
B be the set of people who speak French.
A - B be the set of people who speak English and not French.
B - A be the set of people who speak French and not English.
A ∩ B be the set of people who speak both French and English.
Given,
n(A) = 72
n(B) = 43
n(A ∪ B) = 100
Now, n(A ∩ B) = n(A) + n(B) - n(A ∪ B)
= 72 + 43 - 100
= 115 - 100
= 15
Therefore, Number of persons who speak both French and English = 15
n(A) = n(A - B) + n(A ∩ B) ⇒
n(A - B) = n(A) - n(A ∩ B)
= 72 - 15
= 57
and n(B - A) = n(B) - n(A ∩ B)
= 43 - 15
= 28
Therefore, Number of people speaking English only = 57
Number of people speaking French only = 28
Key Learning:
In this article, we have learnt about the different types of sets as well as formulas to solve questions in a simplified manner.
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