# HW8 - A to B Suppose that g is an injective function from B...

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Name: Math 100 Due Friday, November 30, 2:00 pm Homework 8 1. (2 points) Let f : R R be the function given by the formula: f ( x ) = x 3 - x + 1 , for all x R . Is f injective? Is f surjective? Justify your answers. Feel free to use the intermediate value theorem and basic calculus, if you want. 2. (2 points) Suppose that A , B , and C are sets. Suppose that f is a function from A to B . Suppose that g is a function from B to C . Let h = g f be the function from A to C , deﬁned by: h ( a ) = g ( f ( a )) , for every a A. Prove that if f is surjective and g is surjective, then h is surjective. 3. (2 points) Suppose that A and B are ﬁnite sets. Suppose that f is an injective function from
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Unformatted text preview: A to B . Suppose that g is an injective function from B to A . Prove that f is bijective. 4. (2 points) Let A = { 1 , 2 , 3 , 4 } . A function f from A to A is called non-decreasing if, for all x,y A , if x y , then f ( x ) f ( y ). For example, the following is a non-decreasing function: f (1) = 1 ,f (2) = 3 ,f (3) = 3 ,f (4) = 4 . How many non-decreasing functions are there from A to A ? Justify your answer. (Hint: try a direct counting argument, not a fancy formula!)...
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