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# w5 - f is not surjective then g ◦ f is not surjective 5...

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Math 55 Worksheet Adapted from worksheets by Rob Bayer, Summer 2009. Sets, Functions, Cardinality 1. Determine whether each of the following are injective, surjective, both (bijective), or neither. Prove your answer (a) f : N N f ( x ) = x 2 (b) f : R R f ( x ) = 3 x + 4 (c) f : (0 , 1) (1 , ) f ( x ) = 1 x 2. For a function f : A B , given S A , let f ( S ) := { y B : x S, f ( x ) = y } . Prove that if f is one-to-one, then f ( S T ) = f ( S ) f ( T ) for any S, T A . Then, give an example of a non-injective function f , and sets S, T for which this is not true. 3. Use the definitions of the set complement, union and intersection to prove that A B = A B 4. Decide whether each of the following are true or false. For those that are true, prove it. For those that are false, provide a counterexample. (a) If f and g are injective, then so is f g . (b) If f and g are surjective, then so is f g . (c) If g f is injective, then so is f . (d) If g f is injective, then so is g .
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Unformatted text preview: f is not surjective, then g ◦ f is not surjective. 5. Even when a function f : A → B is not invertible, we will often talk about the inverse image of a set C ⊆ B , which we denote by f-1 ( C ) and which is deﬁned as { x | f ( x ) ∈ C } . Determine each of the following: (a) f-1 ( { September 12 } ) where f :(People) → (days of year), f ( x ) = x ’s birthday. (b) f-1 ( { x | x > 3 } ) where f : R → R ; f ( x ) = x 2 . (c) f-1 ( { 3 } ) where f : P ( N ) → N , f ( S ) = min( S ) (or 0 if S = ∅ ). 6. Prove that if the last digit of n is 3, then n is not a perfect square. 7. Prove that if a ≡ b (mod m) and c ≡ d (mod m) then ac ≡ bd (mod m) Hint: To write your proof. Write down the hypotheses using the deﬁnition of “mod”...
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