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w4 - Math 55 Worksheet Adapted from worksheets by Rob Bayer...

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Math 55 Worksheet Adapted from worksheets by Rob Bayer, Summer 2009. Functions 1. Find the value of 3 i =1 3 j =1 ( i - j ) 2. Determine whether each of the following are injections, surjections, bijections, or none of the three. (a) f : R R , f ( x ) = 3 x - 2 (b) f : R Z , f ( x ) = b x c (c) f : N N , f ( n ) = n + 1 (d) f : R R , f ( x ) = x ( x - 3)( x + 2) 3. Write down the formal definitions of a function , injectivity and surjectivity . These will help with problem ?? . 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 . (e) If f is not surjective, then g f is not surjective. 5. Explain why non-injective functions have no inverse function 6. The Hotel Infinity is rather unique, in that it has infinitly many rooms, one for each natural number. However, due to a recent surge in demand, all the rooms are currently full and people are still coming at a ludicrous rate.
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