Set Theory | General Test Preparation for CUET UG - CUET Commerce PDF Download

What is a Set? 

Set theory is a branch of mathematics that studies collections of objects and how to work with them. A set is a well-defined collection of distinct objects. The objects in a set are called elements or members.

Example: The set of natural numbers:
$N=\{1,2,3,4,\dots \hspace{0.17em}\}N\; =\; \backslash \{1,\; 2,\; 3,\; 4,\; \backslash dots\backslash \}$N={1,2,3,4,…}.
“Well-defined” means anyone can determine whether a given object belongs to the set.

Important Terms Related to Sets

Elements of a Set

• The objects contained by a set are called the elements of the set.
• They are represented using the ∈ symbol, which means “belongs to”.
• For Example:
  In the set of Natural Numbers, 1, 2, 3, etc., are the objects; hence, they are the elements of the set of Natural Numbers.
  We can also say that 1 belongs to set N, and it is represented as 1 ∈ N.

Cardinality of a Set

• The number of elements present in a set is called the Cardinality of a Set (denoted |A| or n(A)). 
• For Example:
  Suppose P is a set of the first five prime numbers given by P = {2, 3, 5, 7, 11}, then the Cardinal Number of set P is 5.
  The Cardinality of Set P is represented by n(P) or |P| = 5.

Representation of Sets

Sets can be represented in two ways:
1. Roster Form or Tabular Form
2. Set Builder Form

1. Roster Form

• In roster form, all the elements of the set are listed, separated by commas and enclosed between curly braces { }. 
• Example: If the set represents all the leap years between the year 1995 and 2015, then it would be described using Roster form as:
  A ={1996,2000,2004,2008,2012}
• Elements are often listed in ascending order by convention, but order does not matter for a set. 
• Also, multiplicity is ignored while representing the sets. E.g. If L represents a set that contains all the letters in the word ADDRESS, the proper Roster form representation would be
• L ={A, D, R, E, S }= {S, E, D, A, R}  
  L≠ {A, D, D, R, E, S, S}

2. Set Builder Form

• In set builder form, all the elements have a common property. This property is not applicable to the objects that do not belong to the set. 
• Example: If set S has all the elements that are even prime numbers, it is represented as:
  S={ x: x is an even prime number}
• where ‘x’ is a symbolic representation that is used to describe the element.
• ‘:’ means ‘such that’
• ‘{}’ means ‘the set of all’
• So, S = { x:x is an even prime number } is read as ‘the set of all x such that x is an even prime number’. The roster form for this set S would be S = {2}. This set contains only one element. Such sets are called singleton/unit sets.

Types of Sets

Set Theory Symbols

Given below are all the symbols that are used in set theory:

Set Operations

1. Union (U):

• The combination of elements from two sets, including duplicates.
• Example:
  If Set A = {1, 3, 5} and Set B = {2, 3, 4}, then A U B = {1, 2, 3, 4, 5}

2. Intersection (∩):

• The collection of elements that are in both sets.
• Example:
  If Set A = {1, 3, 5} and Set B = {2, 3, 4}, then A ∩ B = {3}

3. Difference (\):

• The collection of elements in the first set that are not in the second set.
• Example:
  If Set A = {1, 3, 5} and Set B = {2, 3, 4}, then A \ B = {1, 5}

4. Complement (A’):

• The collection of elements that are not in the first set but are in the universal set.
• Example:
  If Universal Set = {1, 2, 3, 4, 5} and Set A = {2, 4}, then A’ = {1, 3, 5}

5. Cartesian Product (A x B):

• The collection of ordered pairs where the first element comes from the first set and the second element comes from the second set.
• Example:
  If Set A = {1, 2} and Set B = {a, b}, then A x B = {(1, a), (1, b), (2, a), (2, b)}

Properties of Set Operations

Set Theory Formulas

• For the size of the union of two overlapping sets A and B:
  n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
• For the size of the intersection of two overlapping sets A and B:
  n(A ∩ B) = n(A) + n(B) – n(A ∪ B)
• For the size of set A in terms of its overlap with set B:
  n(A) = n(A ∪ B) + n(A ∩ B) – n(B)
• For the size of set B in terms of its overlap with set A:
  n(B) = n(A ∪ B) + n(A ∩ B) – n(A)
• For the size of set A, excluding the elements also in set B:
  n(A – B) = n(A ∪ B) – n(B)
  n(A – B) = n(A) – n(A ∩ B)
• For the size of the union of three sets A, B, and C:
  n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A∩B) – n(B∩C) – n(C∩A) + n(A∩B∩C)
• Disjoint Sets 

Examples

Q1: If U = {a, b, c, d, e, f}, A = {a, b, c}, B = {c, d, e, f}, C = {c, d, e}, find (A ∩ B) ∪ (A ∩ C).
Sol: A ∩ B = {a, b, c} ∩ {c, d, e, f}
A ∩ B = { c }
A ∩ C = { a, b, c } ∩ { c, d, e }
A ∩ C = { c }
∴ (A ∩ B) ∪ (A ∩ C) = { c }

Q2: Give examples of finite sets.
Sol: The examples of finite sets are:
Set of months in a year
Set of days in a week
Set of natural numbers less than 20
Set of integers greater than -2 and less than 3

Q3: If U = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, A = {3, 5, 7, 9, 11} and B = {7, 8, 9, 10, 11}, then find (A – B)′.
Sol: A – B is a set of members which belong to A but do not belong to B
∴ A – B = {3, 5, 7, 9, 11} – {7, 8, 9, 10, 11}
A – B = {3, 5}
According to the formula,
(A − B)′ = U – (A – B)
∴ (A − B)′ = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} – {3, 5}
(A − B)′ = {2, 4, 6, 7, 8, 9, 10, 11}.

Q4: If A and B are two sets such that n(A) = 17, n(B) = 23 and n(A ∪ B) = 38, then find n(A ∩ B).
Sol: We know that n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
⇒ 38 = 17 + 23 – n(A ∩ B)
⇒ n(A ∩ B) = 40 – 38 = 2

Q5: If X = {1, 2, 3, 4, 5}, Y = {4, 5, 6, 7, 8}, and Z = {7, 8, 9, 10, 11}, find (X ∪ Y), (X ∪ Z), (Y ∪ Z), (X ∪ Y ∪ Z), and X ∩ (Y ∪ Z)
Sol: (X ∪ Y) = {1, 2, 3, 4, 5} ∪ {4, 5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}
(X ∪ Z) = {1, 2, 3, 4, 5} ∪ {7, 8, 9, 10, 11} = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}
(Y ∪ Z) = {4, 5, 6, 7, 8} ∪ {7, 8, 9, 10, 11} = {4, 5, 6, 7, 8, 9, 10, 11}
(X ∪ Y ∪ Z) = {1, 2, 3, 4, 5} ∪ {4, 5, 6, 7, 8} ∪ {7, 8, 9, 10, 11} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
X ∩ (Y ∪ Z) = {1, 2, 3, 4, 5} ∩ {4, 5, 6, 7, 8, 9, 10, 11} = {4, 5}