Question

4. Define a function f:N → Z by tof n/2 if n is even 1-(n + 1)/2 if n is odd. f(n) = Show that f is a bijection.

Answer 1

Ans. 4. fin - 2 f(n) = 2 - (ne) if n is even if n is oda To show f is bijection, first we show it one-one let f(n) = f (m2) mina EIN In+1) MAN n is old - anti a M₂ +) - na - n = n2 if To is show one - me f is How onto. ut y = fm)

- 29 sol (n) = 2m il n is add y = -(441) a n--ay-1 = f'(n) = -2n-1 - -an- به لم -mr , if n is old f' is defined for every n ez f is onto. This f is bijection.

Aus. 7 et x= R XR and R be a relation on x. ((a, g, (w, 2)) ER 7 CEFR 1203 soto (is for Reflexive (cang), (49) nax, gay- (ny), (2.9) ER = R is reflexive. (n. Symmetric let (cony), (tw, 2)) ER 2 waca, z=cy - an - w, ya ſea & na ww , y = K2 K ER &- = ((0, 2), (2,9%) & R. - Ris Symmetric. (ii) i t transitive (ings (W, 2)) ER and ((w, 2), (2, 2)) ER Teacher's Signature

claim :- (Conius, (o, ad) ER. . : (Lug), (0, 2)) ((w, 2), (1, 2)) ER 3 wca zocy and - 6,w, quar - Pa (en) , qa alcy po cica , q- acy & P = ln q = k'ya = ((nd), (8,9) ER KE 1R 203 R is transitive. o r is Reflexive, Symmetric and trasitive. so & is an equivalance relation. [ca, os JA = {(pa) : (10.02.1p, a) <R] - {(lt). Paco, q=0.03 = {(0,0)3. 2] - { lary): (eurs, (a, b)) ER] 3 lan b) a = b = c2 - 1(4,2) CeR os? Teacher's Signature [

Date: Notes Ame. 8. let R be an eavivalance relation on X. then {ado = [6] claim of a, b et it be [ada let & & [a]e and given that belade arb again det =) y Rb given that b G [a Je =) 2 је Га), [b Je c [aja