PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions for Mathematics

If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that: 

\[\left( A \cup B \right)' = A' \cap B'\] 

If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:

\[\left( A \cap B \right)' = A'B' .\]

For any two sets A and B, prove that 

 B ⊂ A ∪ B         

For any two sets A and B, prove that 

A ∩ B ⊂ A             

For any two sets A and B, prove that A ⊂ B ⇒ A ∩ B = A 

For any two sets A and B, show that the following statements are equivalent:

(i) \[A \subset B\] 

(ii) \[A \subset B\]=ϕ 

(iii) \[A \cup B = B\]

(iv) \[A \cap B = A .\] 

For three sets A, B and C, show that \[A \cap B = A \cap C\]

For three sets A, B and C, show that \[A \subset B \Rightarrow C - B \subset C - A\] 

For any two sets, prove that: 

\[A \cup \left( A \cap B \right) = A\] 

For any two sets, prove that: 

\[A \cap \left( A \cup B \right) = A\]

Find sets A, B and C such that \[A \cap B, A \cap C \text{ and } B \cap C\]are non-empty sets and\[A \cap B \cap C = \phi\]

For any two sets A and B, prove that: \[A \cap B = \phi \Rightarrow A \subseteq B'\] 

If A and B are sets, then prove that  \[A - B, A \cap B \text{ and } B - A\] are pair wise disjoint. 

Using properties of sets, show that for any two sets A and B,\[\left( A \cup B \right) \cap \left( A \cap B' \right) = A\] 

For any two sets of A and B, prove that: 

\[A' \cup B = U \Rightarrow A \subset B\] 

For any two sets of A and B, prove that: 

\[B' \subset A' \Rightarrow A \subset B\]

Is it true that for any sets A and \[B, P \left( A \right) \cup P \left( B \right) = P \left( A \cup B \right)\]? Justify your answer.

Show that for any sets A and B, A = (A ∩ B) ∪ ( A - B)

Show that for any sets A and B, A ∪ (B – A) = (A ∪ B)