The number of different $$n$$ $$x$$ $$n$$ symmetric matrices with each elements being either $$0$$ or $$1$$ is (Note: power ($$2,$$ $$x$$) is same as $${2^x}$$)

Consider the binary relation: $$S = \left\{ {\left( {x,y} \right)|y = x + 1\,\,and\,\,x,y \in \left\{ {0,1,2,...} \right\}} \right\}$$

The reflexive transitive closure of $$S$$ is

The inclusion of which of the following sets into

S = { {1, 2}, 1, 2, 3}, {1, 3, 5}, {1, 2, 4},
{1, 2, 3, 4, 5} }
Is necessary and sufficient to make S a complete lattice under the partial order defined by set containment?

The following is the incomplete operation table of a 4-element group. The last row of the table is