\(\text { The domain is }[0, \infty)\)
\(\text { The codomain is also }[0, \infty)\)

\(\mathrm{f}(x)=\left\lfloor\begin{array}{lll} x & -x \mid, \mathrm{f}(x) \end{array}=\left\lfloor\begin{array}{ll} x & -x \end{array}\right.\right.\)
\(f(0) =0-0=0 \)
\(f(6.5) =6-6.5=|-0.5|=0.5 \)
\(f(-7.2) =8-7.2=0.8 \)
\(\therefore \text { Range is }[0,1)\)

\(-1 \leq \sin x \leq 1 \)
\(-2 \leq 2 \sin x \leq 2 \)
\(2 \geq-2 \sin x \geq-2 \)
\(-2 \leq-2 \sin x \leq 2 \)
Adding 1,
\(1-2 \leq 1-2 \sin x \leq 1+2 \)
\(-1 \leq 1-2 \sin x \leq 3 \)
\(-1 \geq \frac{1}{1-2 \sin x} \geq \frac{1}{3} \)
\(\frac{1}{3} \leq \frac{1}{1-2 \sin x} \leq-1 \)
\(\therefore \text { Range is }(-\infty,-1] \cup\left[\frac{1}{3}, \infty\right)\)

\(\text { If } 4 \in X \text { then }(4,4) \notin \mathrm{R}\)
R is not reflexive
Symmetric can be easily checked

Let X = (a,b,c)
Then R = Universal relation
= {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)}.
It is transitive

LetA = (a, b),B = (a,b,c)
A \(\times\)B = {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c)}
B \(\times\)A = {(a, a),(a, b), (b, a), (b, b), (c, a),(c, b)}
(A \(\times\)B) ⋂ (B \(\times\)A) = {(a, a), (a,b), (b, a): (b, b)}
= A \(\times\)A

Let A = {1, 2,3,4,8} = {5, 2,3,6}
A and B have two elements in common.
Number of elements common to A \(\times\) B and B x A = 2 \(\times\)2 = 2²
Similarly here we have 17² element in common

\(n[(A \times B) \cup(A \times C)]=n(A) \times n(B \cup C)\)
= 2 \(\times\) 3 = 6

\((A \times B) \cap(A \times C) =A \times(B \cap C) \)
\(n[(A \times B) \cap(A \times C)] =n(A) \times n(B \cap C) \)
\(\Rightarrow 8 =n(A) \times 2 \Rightarrow n(A)=4 \)