Let R be a non-emply relation on a collection of sets defined by $${A^R}\,B $$ if and only if $$A\, \cap \,B\, = \,\phi $$. Then, (pick the true statement)

Let A be the set of all nonsingular matrices over real numbers and let * be the matrix multiplication operator. Then

Some group (G, o) is known to be abelian. Then, which one of the following is true for G?

The transitive closure of the relation
$$\left\{ {\left( {1,2} \right)\left( {2,3} \right)\left( {3,4} \right)\left( {5,4} \right)} \right\}$$
on the set $$A = \left\{ {1,2,3,4,5} \right\}$$ is ________ .