 # 14 let f r r r r defined by f x y x y x y and let g f

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14.Letf:RR!RRdefined byf(x, y) := (x+y, x-y), and letg=ff. Prove thatgis a bijection.
15.LetXbe a finite nonempty subset ofN. Let|X|=n, and assume thatn2. LetW= 2X-{;}. Definea functionq:W!Xsuch that for anyA2W,q(A)is the maximum natural number inA.(a)Prove thatqis surjective.
(b)Prove thatqinjective=)2n-1n.
(c)Prove by ordinary induction onnthat for alln2, n+ 1<2n.
(d)Prove thatqis not injective.
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16.LetXandAbe sets such that; 6=AX, and letf:X!Xbe a function (Warning:the domain andthe codomain are thesameset). Recall the definition of thedirect image ofAthroughf:f(A) ={y2X|9a2A, y=f(a)}Now denotef(A)byB.Bis another subset ofX, so now we can talk about the direct imagef(B); in factwe can writef(f(A))instead off(B). Prove thatf(A)A=)f(f(A))f(A).
17.Consider sequences ofnbits,n1, in which there are exactlyk0’s, wherek < n, and every 0 isimmediately followed by a 1. How many such sequences are there? (Give the answer and an explanationof how you figured it out. No proofs required.)

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Empty set, Finite set
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