Let A = {0,1, 2, 3} and define a relation R on A as follows:
R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R Reflexive? Symmetric? Transitive?

3

(a)

Let f : \(R\rightarrow R\) be defined by \(f(X)=X^{ 2 }+1\) Find the pre-image of
(i) 17
(ii) -3.

3

(a)

Let the function f:\(R\rightarrow R\) to be defined by:
\(f(x)=cosx\) for all \(x\in R\).
Show that 'f' is neither one-one nor onto.

3

(a)

In the set of natural numbers N, define a relation R as follows:
\(\forall n,m\in N,\ nRm\) if on division by 5 each of the integers \(n\) leaves the remainder less than 5 i.e. one of the numbers 0,1,2,3 and 4. Show that R is equivalence relation. Also, obtain the pairwise disjoint subsets determined by R.

3

(a)

Is the binary operation '*' defined on Z (set of integers) by:
\(m*n=m-n+mn\quad for\quad all\quad m,n\in Z\) commutative?

3

(a)