1. Set:

A set is a well-defined and distinct collection of objects and it is denoted by capital letters like $A, B, C . . .$

2. Representation of Sets:

There are two methods of representing a set,

(i) Roster or Tabular form:

We list all the members of the set within braces $\{\}$ and separate by commas.

For example: $A = \{1, 2, 3, 4\}$

(ii) Set-builder form:

We list the property or properties satisfied by all the elements of the sets.

For example: $A = \{x : x \in N, x < 5\}$

3. Types of Sets:

(i) Empty Set:

A set consisting of no element is called the Empty set and is denoted by $ϕ .$

(ii) Singleton Set:

A set consisting of a single element is called a singleton set.

(iii) Finite and infinite Set:

A set consisting of a finite number of elements is called a finite set, otherwise the set is called an infinite set.

(iv) Equal Sets:

Two sets $A$ and $B$ are equal if they have exactly the same elements.

(v) Equivalent Sets:

Two finite sets $A$ and $B$ are said to be equivalent if the number of elements are equal, i.e. $n (A) = n (B)$

(vi) Universal Set:

The set of all elements of all related sets. It is usually denoted by the symbol $U$ .

(vii) Disjoint Sets:

Two sets which do not have any element in common are called Disjoint Sets.

4. Subset & Superset:

(i) Subset:

A set $A$ is said to be a subset of a set $B$ , if every element of $A$ is also an element of $B$ i.e, $A \subseteq B$ , if $x \in A \Rightarrow x \in B$ .

(ii) Superset:

A set $B$ is said to be the Super set of a set $A$ , if every element of $A$ is also an element of $B$ .

5. Some Important Points:

(i) Every set is a subset of itself.

(ii) The empty set is a subset of every set.

(iii) The total number of subsets of a finite set containing $n$ elements is $2^{n}$

6. Intervals as Subsets of $R$ :

If $a, b$ are real numbers such that $a < b,$ then the set

(i) Closed Interval:

$\{x : x \in R$ and $a \leq x \leq b\}$ is called the closed interval $[a, b]$

(ii) Open Interval:

$\{x : x \in R$ and $a < x < b\}$ is called the open interval $(a, b)$

(iii) Semi-open or Semi-closed Interval:

(a) $\{x : x \in R$ and $a \leq x < b\}$ is called the semi-open or semi-closed interval $[a, b) .$

(b) $\{x : x \in R$ and $a < x \leq b\}$ is called the semi-open or semi-closed interval $(a, b] .$

7. Power Set:

The collection of all subsets of a set $A$ is called the power set of $A$ and is denoted by $P (A)$ .

8. Operations of Sets:

(i) Union of two sets:

The union of two sets $A$ and $B$ is the set of all those elements which are either in $A$ or in $B$ or in both and is denoted by $A \cup B .$ Thus, $A \cup B = \{x : x \in A or x \in B\} .$

(ii) Intersection of two sets:

The intersection of two sets $A$ and $B$ is the set of all those elements which are common to both $A$ and $B$ and is denoted by $A \cap B .$

(iii) Difference of sets:

The difference $A - B$ of two sets $A$ and $B$ is the set of all those elements of $A$ which do not belong to $B$ i.e. $A - B = \{x : x \in A and x \notin B\} .$ .Similarly, $B - A = \{x : x \in B and x \notin A\} .$

(iv) Symmetric difference of two sets:

The symmetric difference of two sets $A$ and $B$ is the set $(A - B) \cup (B - A)$ and is denoted by $A Δ B .$

(v) Complement of a set:

The complement of a subset $A$ of universal set $U$ is the set of all those elements of $U$ which are not the elements of $A$ . The complement of $A$ is $A'$ or $A^{c}$ .

9. Laws of Algebra of Sets:

For any three sets $A, B$ and $C$ , we have

(i) $A \cup A = A$ and $A \cap A = A$ (Idempotent laws)

(ii) $A \cup ϕ = A$ and $A \cap U = A$ (Identity laws)

(iii) $A \cup B = B \cup A$ and $A \cap B = B \cap A$ (Commutative laws)

(iv) $(A \cup B) \cup C = A \cup (B \cup C)$ and $(A \cap B) \cap C = A \cap (B \cap C)$ (Associative laws)

(v) $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ (Distributive laws)

(vi) $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$ (De Morgan's laws)

10. Formulae to Solve Practical Problems on Union and Intersection of Two Sets:

If $A, B$ and $C$ are finite sets and $U$ be the finite universal set, then

(i) $n (A \cup B) = n (A) + n (B) - n (A \cap B)$

(ii) $n (A \cup B) = n (A) + n (B) \Leftrightarrow A, B$ are disjoint non-void sets

(iii) $n (A - B) = n (A) - n (A \cap B)$ i.e., $n (A - B) + n (A \cap B) = n (A)$

(iv) $n (A Δ B) = n (B - A) + n (A - B) = n (A) + n (B) - 2 n (A \cap B)$

(v) $n (A \cup B \cup C) = n (A) + n (B) + n (C) - n (A \cap B) - n (B \cap C) - n (C \cap A) + n (A \cap B \cap C)$

(vi) Number of elements in exactly two of sets $A, B$ and $C$ $= n (A \cap B) + n (B \cap C) + n (C \cap A) - 3 n (A \cap B \cap C)$

(vii) Number of elements in exactly one of sets $A, B$ and $C$ $= n (A) + n (B) + n (C) - 2 n (A \cap B) - 2 n (B \cap C) - 2 n (A \cap C) + 3 n (A \cap B \cap C) .$

What is meant by subset?

Important Questions on Sets

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What is meant by subset?