Question

Question4 please

(1). Let f: Z → Z be given by f(x) = x2. Find F-1(D) where (a) D = {2,4,6,8, 10, 12, 14, 16}. (b) D={-9, -4,0, 16, 25}. (c) D is the set of prime numbers. (d) D = {2k|k Ew} (So D is the set of non-negative integer powers of 2). (2). Suppose that A and B are sets, C is a proper subset of A and F: A + B is a 1-1 function. Show that f[C] (the image of C under f) is a proper subset of B. (3). Suppose that A, B, C and D are sets such that An C = 0 and BND = 0. Suppose further that f: A + B and g: C + D are bijections and define the function h: AUC → BUD by: f(x) for x E A h(x) = { 18(x) for x EC (a) Show that h(x) is a bijection (b) Show that h-1: BUD+ AUC is given by: 17-1(y) for y E B h-ly)= { 18-1(y) for y ED (4). Find a bijection from w to N. What is the inverse of your bijection?

Answer 1

(4)

Bijection : A function is bijection if it is both one - one and onto.

One -one : Every element in the range set of function has a unique pre image in domain set.

Onto: Every element in the co-domain set has atleast one pre image in the domain set, i.e. range set = co-domain set.

We prove these properties for the chosen function f(x).

And the find its inverse.

I (4.) So find: Bijection from W-N w = set of whole numbers Solutions - fake ((x) = x+ 1 N = set of natural = " numbers + XEW Then well defined: (1) " for every XEW, we have a unique image xt1 in on riit some x, y EW one-one: Let f(u) = fly) for 5) x+2 = 4+1 => x =y each element in the range set a unique pre-image in domain 8 is one-one. in has (11.) canto : For every yeN we have an element XE w sto f(x) = y =) set1 = y = (n = 47 So every element in the co-domain has a pre-image in domain. ie Range f = Co domain f a) of is onto Hence of is a bijection from W AN Now cs Scanned be to find ft N the W in verse by of of fr(y) = x & YEN CamScanner

= y = f(x) =) y = S(+1 = x=yt i f (y) = x=yt # yein Let ftly)= gly) then inverse of f" =) : INJW Scanned with by Igly) = y1) & YEN ans CS CamScanner