Relations and Functions (Class 12) · Question 1

Let N denote the set of all positive integers. Consider the sets A=\1,2,3,4,5\ and B=\1,2,3,4,5,6,7\. Let S be the set of all functions f:A B such that f(2) 2 and f(4) 4. Consider the set T=\f S: there exists a function g:B N such that g(f(x))=2x for all x A\. Then the number of elements in the set T is .

Let N denote the set of all natural numbers, and Z denote the set of all integers. Consider the functions f: N Z and g: Z N defined by f(n) = cases (n+1)/2 & if n is odd, \ (4-n)/2 & if n is even, cases and g(n) = cases 3 + 2n & if n 0, \ -2n & if n < 0. cases Define (g f)(n) = g(f(n)) for all n N, and (f g)(n) = f(g(n)) for all n Z. Then which of the following statements is (are) TRUE?

1. g f is NOT one-one and g f is NOT onto
2. f g is NOT one-one but f g is onto
3. g is one-one and g is onto
4. f is NOT one-one but f is onto

Let the set of all relations R on the set \a, b, c, d, e, f\, such that R is reflexive and symmetric, and R contains exactly 10 elements, be denoted by S. Then the number of elements in S is .

Let f: R R be a function such that f(x+y)=f(x)+f(y) for all x,y R, and g: R (0, ) be a function such that g(x+y)=g(x)g(y) for all x,y R. If f ( -35 )=12 and g ( -13 )=2, then the value of (f ( 14 )+g(-2)-8 )g(0) is .

If n(X)=mC6, then the value of m is .

If the value of n(Y)+n(Z) is k2, then |k| is .