Prove the following laws of the algebra of sets.(use membership tables or Venn diagram)

  (a) Indempotentlaws (i.e.(i)A∪A =A

(b) Associativelaws

(i) (A∪B)∪C=A∪(B∪C)

(c) Commutative laws

(i) A∪B=B∪A

(d) Distributive laws

(ii) A∩A =A) (ii)(A∩B)∩C=A∩(B∩C) (ii) A∩B =B∩A

(i) A∪(B∩C)=(A∪B)∩(A∪C) (ii)A∩(B∪C)=(A∩B)∪(A∩C)

(e) Identity law

(i) A∪φ=A (ii)A∩U = A (iii)A∪U=U (iv)A∩φ = φ

Let's prove the given laws of the algebra of sets using both membership tables and Venn diagrams.

### (a) Idempotent Laws
- **(i) \( A \cup A = A \)**

#### Membership Table:
\[
\begin{array}{c|c|c}
x \in A & x \in A \cup A & x \in A \\
\hline
\text{True} & \text{True} & \text{True} \\
\text{False} & \text{False} & \text{False} \\
\end{array}
\]
The table shows that \( x \) belongs to \( A \cup A \) if and only if it belongs to \( A \). Therefore, \( A \cup A = A \).

#### Venn Diagram:
In the Venn diagram, the area representing \( A \) does not change when taking the union of \( A \) with itself. Hence, \( A \cup A = A \).

- **(ii) \( A \cap A = A \)**

#### Membership Table:
\[
\begin{array}{c|c|c}
x \in A & x \in A \cap A & x \in A \\
\hline
\text{True} & \text{True} & \text{True} \\
\text{False} & \text{False} & \text{False} \\
\end{array}
\]
The table shows that \( x \) belongs to \( A \cap A \) if and only if it belongs to \( A \). Therefore, \( A \cap A = A \).

#### Venn Diagram:
In the Venn diagram, the area representing \( A \) does not change when taking the intersection of \( A \) with itself. Hence, \( A \cap A = A \).

### (b) Associative Laws
- **(i) \( (A \cup B) \cup C = A \cup (B \cup C) \)**

#### Membership Table:
\[
\begin{array}{c|c|c|c|c}
x \in A & x \in B & x \in C & x \in (A \cup B) \cup C & x \in A \cup (B \cup C) \\
\hline
\text{True} & \text{True} & \text{True} & \text{True} & \text{True} \\
\text{True} & \text{True} & \text{False} & \text{True} & \text{True} \\
\text{True} & \text{False} & \text{True} & \text{True} & \text{True} \\
\text{True} & \text{False} & \text{False} & \text{True} & \text{True} \\
\text{False} & \text{True} & \text{True} & \text{True} & \text{True} \\
\text{False} & \text{True} & \text{False} & \text{True} & \text{True} \\
\text{False} & \text{False} & \text{True} & \text{True} & \text{True} \\
\text{False} & \text{False} & \text{False} & \text{False} & \text{False} \\
\end{array}
\]
The table shows that \( x \) belongs to \( (A \cup B) \cup C \) if and only if it belongs to \( A \cup (B \cup C) \). Therefore, \( (A \cup B) \cup C = A \cup (B \cup C) \).

#### Venn Diagram:
The area covered by the union of the sets does not change, regardless of the grouping. The Venn diagram would illustrate the same area for both expressions, confirming that \( (A \cup B) \cup C = A \cup (B \cup C) \).

- **(ii) \( (A \cap B) \cap C = A \cap (B \cap C) \)**

#### Membership Table:
\[
\begin{array}{c|c|c|c|c}
x \in A & x \in B & x \in C & x \in (A \cap B) \cap C & x \in A \cap (B \cap C) \\
\hline
\text{True} & \text{True} & \text{True} & \text{True} & \text{True} \\
\text{True} & \text{True} & \text{False} & \text{False} & \text{False} \\
\text{True} & \text{False} & \text{True} & \text{False} & \text{False} \\
\text{True} & \text{False} & \text{False} & \text{False} & \text{False} \\
\text{False} & \text{True} & \text{True} & \text{False} & \text{False} \\
\text{False} & \text{True} & \text{False} & \text{False} & \text{False} \\
\text{False} & \text{False} & \text{True} & \text{False} & \text{False} \\
\text{False} & \text{False} & \text{False} & \text{False} & \text{False} \\
\end{array}
\]
The table shows that \( x \) belongs to \( (A \cap B

Learn how to prove the laws of the algebra of sets, including Idempotent, Associative, Commutative, Distributive, and Identity laws. Understand these fundamental concepts with detailed membership tables and Venn diagrams.

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