GATE Questions & Answers of Sets

Let $ U=\{1,2,\;...\;,\;n\}. $ Let $ A=\{(x,\;X)\;\vert x\in X,\;X\subseteq U\}. $ Consider the following two statements on |A|.

I.  $ \vert A\vert=n2^{n-1} $

II. $ \vert A\vert={\textstyle\sum_{k=1}^n}k\begin{pmatrix}n\\k\end{pmatrix} $

Which of the above statements is/are TRUE?

•  (A)  Only I
•  (B)  Only II
•  (C)  Both I and II
•  (D)  Neither I nor II

Let $G$ be an arbitrary group. Consider the following relations on $G$:

$ R_1:\forall a,\;b\in G,\;a\;R_1b $ if and only if $ \exists g\in G $ such that $ a=g^{-1}bg $

$ R_2:\forall a,\;b\in G,\;a\;R_2b $ if and only if $ a=b^{-1} $ 

Which of the above is/are equivalence relation/relations?

•  (A)  $ R_1\;\mathrm{and}\;R_2 $
•  (B)  $ R_1\;\mathrm{only} $
•  (C)  $ R_2\;\mathrm{only} $
•  (D)  $ \mathrm{Neither}\;R_1\;\mathrm{nor}\;R_2 $

A binary operation $\oplus$ on a set of integers is defined as $x\oplus y=x^2+y^2$ . Which one of the following statements is TRUE about $\oplus$?

•  (A) Commutative but not associative
•  (B) Both commutative and associative
•  (C) Associative but not commutative
•  (D) Neither commutative nor associative