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 f1 ( C ) and which is deﬁned as { x  f ( x ) ∈ C } . Determine each of the following: (a) f1 ( { September 12 } ) where f :(People) → (days of year), f ( x ) = x ’s birthday. (b) f1 ( { x  x > 3 } ) where f : R → R ; f ( x ) = x 2 . (c) f1 ( { 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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 Summer '09
 RobBayer
 Math, Sets, Inverse function, Rob Bayer

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