Question

Consider the following statements.
I. If $A\cap B=\varphi$, then either $A=\varphi$ or $B=\varphi$.
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 $A\cap B=\varphi$, then it is not necessary that $A=\varphi$ or $B=\varphi$.
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 $A\times C\subseteq B\times D$.