wwwmathucsdeduasa x 206 unread elahel 337 x Chegg Study Gu

www.math.ucsd.edu/~asa x (206 unread) _ elahel 337 x Chegg Study | Guided So × -)C Dwww.math.ucsd.edu/~asalehig/math109-f-15-PracticeFinal.pdf I Injection, surjection, bijection. I. Let f : X Y and g : Y Z be two functions. Prove the following a) If g o f is injective, then f is injective. (b) If g o f is surjective, then g is surjective. (c) If g of is a bijection, then the restriction gl1m(f) : Im (f) X of g to the image of f is a bijection (d) If f and g are injective, then gof is injective e) If f and g are surjective, then go f is surjective f) If f and g are bijections, then go f is a bijection 2. Let f : X Y be a function. Prove the following (a) There is a function g : Y X such that go f = 1x if and only if f in injective (b) There is a function g : Y X such that fog = 1Y if and only if f is surjective. (c) f is invertible if and only if f is a bijection (d) If f is invertible, then there is a unique function g : Y X such that go f = 1x and fog-1Y (e) If f is invertible, then its inverse f-1 is a bijection. 3. Determine if the following functions are injective, surjective, or bijective. (a) f : z z, f(x)-x-1 if x is odd, and f(x) = x + 1 if x is even. (This is from Professor Popescu\'s exam b) Let X be a non-empty set, and f : P(X) {g| g : X {0,1}), f(A) := 1A where 1A : X {0,1} is the characteristic function of A, i.e. 1A(z) = 1 ifa E A, and 1A(2) = 0 if 2:21 PM 12/7/2015 Search the web and Windows PTFP10Cli... final exam... PTFP11Qu... www.mat...

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