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487 d l b \ g u _ f _ j u i h \ u y \ e _ g b x g Z j m r _ g b c b e b k m s _ k l \ m x s b o • • •. Check each step in the formula What trig formulae did we use?. F(x) is a function in x only You can draw this in the Cartesian plane in R 2 f(x,y) is function in x and y If you draw this in R 3, the function will lie in the xyplane f(x,y,z) is a function in x,y and z.

MATH 052 INTRODUCTION TO PROOFS HOMEWORK #26 Problem 337 Suppose f A !B and g B !C are functions (a) Show that if g f is injective then f is injective. JOEY HYSTERIC b W C q X e b N T V c/ J b g \ i z C g/ F n j w 邱 Ƃ ł ܂ B z i ꕔ n j p ܂ B ZOZOTOWN JOEY HYSTERIC i W C q X e b N j T V c/ J b g \ i z C g n j ȂǖL x Ɏ 葵 t @ b V ʔ̃T C g ł B { _ n AV l b N T V c ȂǁA ԃA C e ŐV g h A C e ܂ŃI C ł w ܂ B V A C e ג I. SPINNS b X s Y ̃L b v i z C g/ F n j w 邱 Ƃ ł ܂ B z i ꕔ n j p ܂ B ZOZOTOWN SPINNS i X s Y j ̃L b v i z C g n j ȂǖL x Ɏ 葵 t @ b V ʔ̃T C g ł B x X { L b v ⃏ N L b v ȂǁA ԃA C e ŐV g h A C e ܂ŃI C ł w ܂ B V A C e ג I.

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MATH 052 INTRODUCTION TO PROOFS HOMEWORK #26 Problem 337 Suppose f A !B and g B !C are functions (a) Show that if g f is injective then f is injective. D^gigV ^ c ^gedac^X =d\^_ eaVc Zaå gXd_ \^c^, madX` \^Xh, ib^fVh ^ ^Zh X VZ Mb c bc, shd c^bcåh jV`h, mhd i =dYV Wqa gdXfnccq_ eaVc. 3 EQUIVALENCE RELATIONS 34 and since each of x−x0 and y−y0 is an integer (by definition of R), (x−x0)(y−y0) is an integer Thus, (xy)R(x0 y0) Exercise 321 In the example above, show that it is possible to have xRx0 and yRy0, but (xy)˚R(x0y0) Exercise 322.

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0 1 2 3 4 5 6 7 8 9;. CR!XJFG0QA3ZDHQRCG5GZ2SR}„LR}„‹BOOKMOBI ™Í ¸*” 0J 8k @ H® Pý Y aX i« qÍ z ‚/ ‰ô ’ š" £ ªg"²µ$ºï&¸(Ë *Ò«,ÚîãA0ë. First solution We have x = y^a => log(x) = alog(y) => a = log(x)/log(y) (1) y = z^b => log(y) = blog(z) => b = log(y)/log(z) (2) z = x^c => log(z) = clog(x.

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