(a) $$S = \left\{ { < 1,2 > ,\, < 2,1 > } \right\}$$ is binary relation on set $$A = \left\{ {1,2,3} \right\}$$. Is it irreflexive?
Add the minimumnumber of ordered pairs to $$S$$ to make it an $$\,\,\,\,\,$$equivalence relation. Give the modified $$S$$.

(b) Let $$S = \left\{ {a,\,\,b} \right\}\,\,\,\,$$ and let ▢ $$S$$ be the power set of $$S$$. Consider the binary relation $$'\underline \subset $$ (set inclusion)' on ▢ $$S$$. Draw the Hasse diagram corresponding to the lattice (▢$$(S)$$, $$\underline \subset $$)

A multiset is an unordered collection of elements where elements may repeat ay number of times. The size of a multiset is the number of elements in it counting repetitions.

(a) what is the number of multisets of size 4 that can be constructed from n distinct elements so that at least one element occurs exactly twice?
(b) How many multisets can be constructed from n distinct elements?

Let $$S = \left\{ {0,1,2,3,4,5,6,7} \right\}$$ and $$ \otimes $$ denote multiplication modulo $$8$$, that is, $$x \otimes y = \left( {xy} \right)$$ mod $$8$$

(a) Prove that $$\left( {0,\,1,\, \otimes } \right)$$ is not a group.
(b) Write $$3$$ distinct groups $$\left( {G,\,\, \otimes } \right)$$ where $$G \subset s$$ and $$G$$ has $$2$$ $$\,\,\,\,\,\,$$elements.

Let $${G_1}$$ and $${G_2}$$ be subgroups of a group $$G$$.
(a) Show that $${G_1}\, \cap \,{G_2}$$ is also a subgroup of $$G$$.
(b) $${\rm I}$$s $${G_1}\, \cup \,{G_2}$$ always a subgroup of $$G$$?