Consider the following two binary relations on the set A = {a, b, c} :
R₁ = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and
R₂ = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}.
Then :

Let P = {$$\theta $$ : sin$$\theta $$ $$-$$ cos$$\theta $$ = $$\sqrt 2 \,\cos \theta $$}

and Q = {$$\theta $$ : sin$$\theta $$ + cos$$\theta $$ = $$\sqrt 2 \,\sin \theta $$} be two sets. Then

Let A and B be two sets containing four and two elements respectively. Then, the number of subsets of the set A $\times$ B , each having atleast three elements are

Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can be formed such that Y $$ \subseteq $$ X, Z $$ \subseteq $$ X and Y $$ \cap $$ Z is empty, is :