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

R₁: ∀a,b ∈ G, aR₁b if and only if ∃g ∈ G such that a = g^-1bg

R₂: ∀a,b ∈ G, aR₂b if and only if a = b^-1

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

Let U = {1, 2 ,..., n}. Let A = {(x, X) | x ∈ X, X ⊆ U}. Consider the following two statements on |A|.

I. |A| = n2^n–1

II. |A| = $$\sum\limits_{k = 1}^n {k\left( {\matrix{ n \cr k \cr } } \right)} $$

Which of the above statements is/are TRUE?

Suppose $$𝑈$$ is the power set of the set $$S = \left\{ {1,2,3,4,5,6,} \right\}$$. For any $$T \in U,$$ let $$\left| T \right|$$ denote the number of elements in $$𝑇$$ and $$T'$$ denote the complement of $$𝑇.$$ For any $$T,R \in U,$$ let $$T\backslash R$$ be the set of all elements in $$𝑇$$ which are not in $$𝑅.$$ Which one of the following is true?

For a set A, the power set of A is denoted by 2^A. If A = {5, {6}, {7}}, which of the following options are TRUE?

I. $$\phi \in {2^A}$$
II. $$\phi \subseteq {2^A}$$
III. $$\left\{ {5,\left\{ 6 \right\}} \right\} \in {2^A}$$
IV. $$\left\{ {5,\left\{ 6 \right\}} \right\} \subseteq {2^A}$$