4. Two marks exercise problems II

1. If \(U = \{x : x \in \mathbb{N}, x \le 10\}\), \(A = \{2, 3, 4, 8, 10\}\) and \(B = \{1, 2, 5, 8, 10\}\), then verify that \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\).

 

Proof:

 

LHS:

 

\(n(A \cup B) = \)

 

RHS:

 

\(n(A) = \)

 

\(n(B) = \)

 

\(n(A \cap B) = \)

 

\(n(A) + n(B) - n(A \cap B)\) \(=\)

 

 

Hence, proved.

 

 

2. If \(U = \{1, 2, 3, .., 10\}\), \(P = \{3, 4, 5, 6\}\) and \(Q = \{x : x \in \mathbb{N}, x < 5\}\), then verify that \(n(Q - P) = n(Q) - n(P \cap Q)\).

 

Proof:

 

LHS:

 

\(n(Q - P) = \)

 

RHS:

 

\(n(Q) = \) 

 

\(n(P \cap Q) = \)

 

\(n(Q) - n(P \cap Q)\) \(=\)