Question

Consider the following statements.
I. If $A \cap B = \phi$, then either $A = \phi$ or $B = \phi$.
II. For $a \ne b$, $\{a, b\} = \{b, a\}$ and $(a, b) \ne (b, a)$.
III. If $A \subseteq B$, then $A \times A \subseteq (A \times B) \cap (B \times A)$.
IV. If $A \subseteq B$ and $C \subseteq D$, then $A \times C \subseteq B \times D$.
Which of these is/are correct?

• A. I, II and III
• B. Both I and IV
• C. Only IV
• D. II, III and IV

Answer 1

Answer: (D)

I. If $A \cap B = \phi$, then it is not necessary that $A = \phi$ or $B = \phi$.
II. It is true $\{a, b\} = \{b, a\}$ and $(a, b) \ne (b, a)$.
III. By properties of cartesian product
If $A \subseteq B$, then $A \times A \subseteq (A \times B) \cap (B \times A)$.
IV. If $A \subseteq B$ and $C \subseteq D$, then $A \times C \subseteq B \times D$.