Let X be a set and 2$$^X$$ denote the powerset of X. Define a binary operation $$\Delta$$ on 2$$^X$$ as follows:

$$A\Delta B=(A-B)\cup(B-A)$$.

Let $$H=(2^X,\Delta)$$. Which of the following statements about H is/are correct?

Consider a random experiment where two fair coins are tossed. Let A be the event that denotes HEAD on both the throws, B be the event that denotes HEAD on the first throw, and C be the event that denotes HEAD on the second throw. Which of the following statements is/are TRUE?

Let G be a simple, finite, undirected graph with vertex set {$$v_1,...,v_n$$}. Let $$\Delta(G)$$ denote the maximum degree of G and let N = {1, 2, ...} denote the set of all possible colors. Color the vertices of G using the following greedy strategy:

for $$i=1,....,n$$

color($$v_i)$$ $$\leftarrow$$ min{$$j\in N$$ : no neighbour of $$v_i$$ is colored $$j$$}

Which of the following statements is/are TRUE?

Let $$U = \{ 1,2,3\} $$. Let 2$$^U$$ denote the powerset of U. Consider an undirected graph G whose vertex set is 2$$^U$$. For any $$A,B \in {2^U},(A,B)$$ is an edge in G if and only if (i) $$A \ne B$$, and (ii) either $$A \supseteq B$$ or $$B \supseteq A$$. For any vertex A in G, the set of all possible orderings in which the vertices of G can be visited in a Breadth First Search (BFS) starting from A is denoted by B(A).

If $$\phi$$ denotes the empty set, then the cardinality of B($$\phi$$) is ___________