Laws of Algebra of sets (Properties of sets):

• Commutative law

$$A \cup B=B \cup A $$

$$ A \cap B=B \cap A$$

• Associative law

$$(A \cup B) \cup C=A \cup(B \cup C) $$ $$ (A \cap B) \cap C=A \cap(B \cap C)$$

• Distributive law $$A \cup(B \cap C)=(A \cup B) \cap(A \cup C) $$ $$ A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$$

• De-morgan law $$(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime} $$ $$ (A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$$

• Identity law $$A \cap U=A $$ $$ A \cup \phi=A$$

• Complement law

$$A \cup A^{\prime}=U$$ $$ A \cap A^{\prime}=\phi$$ $$ \left(A^{\prime}\right)^{\prime}=A$$

• Idempotent law $$A \cap A=A$$ $$ A \cup A=A$$

Some important results on number of elements in sets:

PYQ-2023-Sequence_and_Series-Q20

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

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

• $$\quad n(A-B)=n(A)-n(A \cap B)$$

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

• Number of elements in exactly two of the sets $A, B, C$

$$n(A \cap B)+n(B \cap C)+n(C \cap A)-3 n(A \cap B \cap C)$$

• Number of elements in exactly one of the sets $A, B, C$ $$ n(A)+n(B)+n(C)-2 n(A \cap B)-2 n(B \cap C)-2 n(A \cap C) +3 n(A \cap B \cap C) $$

• If A has n elements, then P(A) has $2^n$ elements

• The total number of subsets of a finite set containing n elements is $2^n$

• Number of proper subsets of A, containing n elements is $2^n - 1$

• Number of non-empty subsets of A, containing n elements is $2^n - 1$

Types of relations :

PYQ-2023-Sequence_and_Series-Q21

• Void relation

  • Let $\mathrm{A}$ be a set. Then $\phi \subseteq \mathrm{A} \times \mathrm{A}$ and so it is a relation on $A$. This relation is called the void or empty relation on $A$.

• Universal relation

• Let $A$ be a set. Then $A \times A \subseteq A \times A$ and so it is a relation on $A$. This relation is called the universal relation on $A$.

• Identity relation

  • Let $A$ be a set. Then the relation $I_A={(a, a): a \in A }$ on $A$ is called the identity relation on $\mathrm{A}$.

  • In other words, a relation $\mathrm{I}_{\mathrm{A}}$ on $\mathrm{A}$ is called the identity relation if every element of $A$ is related to itself only.

• Reflexive relation PYQ-2023-Sets_And_Relation-Q1 PYQ-2023-Sets_And_Relation-Q3 PYQ-2023-Sets_And_Relation-Q5 PYQ-2023-Sets_And_Relation-Q7

  A relation $R$ on a set $A$ is said to be reflexive if every element of $A$ is related to itself. Thus, $R$ on a set $A$ is not reflexive if there exists an element $a \in A$ such that $(a, a) \notin R$.

• Reflexive Relation Formula

  $$ N = 2^{n^2 - n} $$

  where N is the number of Reflexive relations and n is the number of items in the set, gives the number of reflexive relations on a set with ‘n’ elements.

• Note

  Every identity relation is reflexive but every reflexive relation in not identity.

• Symmetric relation: PYQ-2023-Sets_And_Relation-Q1 PYQ-2023-Sets_And_Relation-Q3 PYQ-2023-Sets_And_Relation-Q4 PYQ-2023-Sets_And_Relation-Q5 PYQ-2023-Sets_And_Relation-Q6 PYQ-2023-Sets_And_Relation-Q7

  A relation $R$ on a set $A$ is said to be a symmetric relation iff $(a, b) \in R \Rightarrow(b, a) \in R$ for all $a, b \in A . \quad$ i.e. $a R b \Rightarrow b R$ for all $a, b \in A$.

• Symmetric Relation Formula

  $$ N = 2^{\frac{n(n+1)}{2}} $$

  where N is the number of symmetric relations and n is the number of items in the set, gives the number of symmetric relations on a set with ‘n’ elements.

• Transitive relation: PYQ-2023-Sets_And_Relation-Q1 PYQ-2023-Sets_And_Relation-Q3 PYQ-2023-Sets_And_Relation-Q4 PYQ-2023-Sets_And_Relation-Q5 PYQ-2023-Sets_And_Relation-Q6 PYQ-2023-Sets_And_Relation-Q7

  Let $A$ be any set. $A$ relation $R$ on $A$ is said to be a transitive relation

  iff $(a, b) \in R$ and $(b, c) \in R \Rightarrow(a, c) \in R$ for all $a, b, c \in A$ i.e. $a R b$ and $b R c \Rightarrow a R c \quad$ for all $a, b, c \in A$

• Equivalence relation: PYQ-2023-Sets_And_Relation-Q2 PYQ-2023-Sets_And_Relation-Q3 PYQ-2023-Sets_And_Relation-Q7 PYQ-2023-Sets_And_Relation-Q8

  A relation $R$ on a set $A$ is said to be an equivalence relation on $A$ iff

  • $\quad$ it is reflexive i.e. (a, a) $\in R$ for all $a \in A$

  • $\quad$ it is symmetric i.e. (a, b) $\in R \Rightarrow(b, a) \in R$ for all $a, b \in A$

  • $\quad$ it is transitive i.e. (a, b) $\in R$ and (b, c) $\in R \Rightarrow(a, c) \in R$ for all $a, b \in A$

  • $\quad$ Number of relation from A to A which are both reflexive and symmetric is $2^{\frac{n^2 - n}{2}}$