Basic Concepts of Set Theory | Types of Sets | Solved Examples

Set:  A set is a well-defined collection of objects or elements. Each element in a set is unique.

Usually but not necessarily a set is denoted by a capital letter e.g. A, B, … U, V etc. and the elements are enclosed between brackets { }, denoted by small letters a , b , ….x , y etc.

For example:

A = Set of all small English alphabets

= {a , b , c ,… , x , y , z}

B = Set of all positive integers less than or equal to 10

= {1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10}

R = Set of real numbers = {x: − ∞ < x < ∞}

The elements of a set can be discrete (e.g. set of all English alphabets) or continuous (e.g. set of real numbers).

The set may contain finite or infinite number of elements. A set may contain no elements and such a set is called Void set or Null set or empty set and is denoted by φ (phi).

The number of elements of a set A is denoted as n(A) and hence n( φ) = 0 as it contains no element.

Union of Sets:

Union of two or more sets is the set of all elements that belong to any of these sets. The symbol used for union of sets is ‘ ∪ ‘

i.e. A∪B = Union of set A and set B

= {x: x ∈ A or x∈B (or both)}

Example:

If A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then A∪B∪C = {1, 2, 3, 4, 5, 6, 8}

Intersection of Sets:

It is the set of all the elements, which are common to all the sets.
The symbol used for
intersection of sets is ‘ ∩ ‘.

i.e. A∩B = {x: x ∈ A and x∈ B}

Example:
If A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then A ∩ B ∩ C = { 2 }
Remember that n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Difference of two sets:

The difference of set A to B denoted, as A – B is the set of those elements that are in the set A but not in the set B

i.e. A – B = {x: x ∈ A and x ∉ B}.
Similarly B − A = {x: x ∈ B and x ∉ A}.

In general A − B ≠ B − A.
Example:
If A = {a, b, c, d} and B = {b , c , e , f}

then A − B = {a, d} and B − A = {e , f}

Subset of a Set:

A set A is said to be a subset of the set B if each element of the set A is also the element of the set B. The symbol used is ‘ ⊆ ‘

i.e. A ⊆ B <=> (x ∈ A

=> x ∈ B)

Each set is a subset of its own set. Also a void set is a subset of any set.

If there is at least one element in B which does not belong to the set A, then A is a proper subset of set B and is denoted as A ⊂ B.

If set B has n elements then total number of subsets of set B is 2n

Example:

If A = {a, b, c, d} and B = {b, c, d} then B ⊂ A or equivalently A ⊃ B (i.e A is a super set of B).

Equality of Two Sets:

Sets A and B are said to be equal if A ⊆ B and B ⊆ A and we write A = B.

Universal Set:

As the name implies, it is a set with collection of all the elements and is usually denoted by U.

e.g. set of real numbers R is a universal set whereas a set A ={x: x ≤ 3} is not a universal set as it does not contain the set of real numbers x >3.

Once the universal set is known, one can define the Complementary set of a set as the set of all the elements of the universal set which do not belong to that set.
e.g. If A = {x: x ≤ 3} then A˜ = complimentary set of A = {x: x > 3}.

Hence we can say that A ∪ A˜ = U i.e. Union of a set and its complimentary is always the Universal set and A ∩ A˜ = φ

i.e intersection of the set and its complimentary is always a void set.

Cartesian product of Sets:

The Cartesian product (also known as the cross product) of two sets A and B , denoted by A x B (in the same order) is the set of all ordered pairs (x , y) such that x ∈ A and y ∈ B.

What we mean by ordered pair is that the pair (a, b) is not the same pair as (b, a) unless a = b. It implies that

A × B ≠ B × A in general. Also if A contains m elements and B contains n elements then A x B contains m x n elements.

Similarly we can define A × A = {(x, y); x ∈ A and y ∈ A}. We can also define cartesian product of more than two sets.

e.g. A1 × A2 × A3  . . . . × An = {(a1, a2, . . . , an) : a1 ∈ A1 , a2 ∈ A2, . . . , an ∈ An}

Solved Example: Sixty five percent of children in a sport club play football , 70 percent play volley-ball and 75 per-cent play basket ball. What is the smallest percentage of children playing all the three games?

Solution: Out of 100 children,

Number of children who do not play football = 100 − 65 = 35

Number of children who do not play volley-ball = 100 − 70 = 30

Number of children who do not play basket ball = 100 − 75 = 25

So, the maximum number of children who do not play at least one game

= 35 + 30 + 25 = 90.

Thus the minimum number of children who play all the three games = 100 − 90 = 10

Hence the smallest percentage of children playing all the three games = 10 %.

Note: The greatest percentage of children playing all the three games = min(65% , 70% , 75%} = 65%

Exercise :
(i) In a small sweet shop people usually buy either one cake or one box of chocolate. One day the shop sold 57 cakes and 36 boxes of chocolates. How many customers were there that day if 12 people bought both a cake and a box of chocolates ?

(ii) A survey shows that 63% of the Americans like cheese whereas 76 % like apples. If x% of the Americans like both cheese and apples, find the values of x.