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MA10209 – Week 2 Tutorial B3/B4, Andrew Kennedy

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Top Tips (response to sheet 1) Make sure you answer the question - requires a yes/no as well as an explanation. Explain your answers in words. Dont rely on diagrams. Be careful not to miss parts of the question accidentally.

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Top Tips (response to sheet 1) Equals means equals. Dont set functions equal to real numbers is a nonsense statement! Consider the difference between infinitely many maps and one map taking infinitely many values. For example, is a single map, but is infinitely many maps.

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Top Tips (response to sheet 1) Know when to use a proof and when to use a counterexample. A counter-example shows a statement doesnt hold, All prime numbers are odd. but sometimes proving the converse is easier. There are infinitely many even prime numbers. A proof is required to show a statement holds. There is exactly one even prime number. Try not to include irrelevant information in your answer.

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Functions/Maps: definition What do we need for a function/map?

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Functions/Maps: definition What do we need for a function/map? Domain Codomain Rule which assigns to each element of the domain a single element of the codomain

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Functions/Maps: properties What do the following special properties mean? InjectiveSurjective Bijective

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What do the following special properties mean? InjectiveSurjective Bijective Functions/Maps: properties

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What do the following special properties mean? InjectiveSurjective Bijective Functions/Maps: properties

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Functions/Maps: examples Are the following (a) injective, (b) surjective, (c) bijective?

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Functions/Maps: examples Is the following (a) injective, (b) surjective, (c) bijective? Injective?

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Functions/Maps: examples Is the following (a) injective, (b) surjective, (c) bijective? Surjective?

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Functions/Maps: examples Is the following (a) injective, (b) surjective, (c) bijective? Bijective?

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Functions/Maps: examples Is the following (a) injective, (b) surjective, (c) bijective? Injective?

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Functions/Maps: examples Is the following (a) injective, (b) surjective, (c) bijective? Surjective?

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Creating bijections Give a bijection between the following two sets:

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Creating bijections Give a bijection between the following two sets: One possible answer: Why is this a bijection?

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Creating bijections Give a bijection between the following two sets: One possible answer: Why is this a bijection?

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Creating bijections Give a bijection between the following two sets: One possible answer: Why is this a bijection?

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Counting maps a) How many maps are there from A to B? b) How many injective maps are there from A to B? c) How many surjective maps are there from A to B? d) How many maps are there from B to A? e) How many injective maps are there from B to A? f) How many surjective maps are there from B to A?

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960806960806 Counting maps a) How many maps are there from A to B? b) How many injective maps are there from A to B? c) How many surjective maps are there from A to B? d) How many maps are there from B to A? e) How many injective maps are there from B to A? f) How many surjective maps are there from B to A?

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Exercise Sheet 2 - overview Q1 – lots of problems. If its not a map, clearly state whats wrong with it. (b) consider. Q2 – if the property holds, prove it does. If it doesnt, find a counter-example (c) & (d) (e) explain! (f) consider graph of. Q3 – considering smaller examples may help find the solution. Remember to justify answers.

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Exercise Sheet 2 - overview Q4 – select the right maps, and everything is reasonably obvious (hopefully) Q5 – remember how to prove injectivity and surjectivity. Use counterexamples if theyre not true. Note: it is a good idea to try to find counterexamples on small sets. This will help you think about exactly where it breaks.

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Exercise Sheet 2 - overview Q6 (a) & (b) use question 5 (c) note that. Q7 (a) fix for some. Show that this is enough to identify the map by using induction (remember youre working on all of ). This give one direction: You also need to show the other.

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Exercise Sheet 2 - overview Q8 – quite tricky. Its designed to get you thinking about the necessary properties of a bijection, and about creative ways to create them. (c) Find a bijection between and. From there, previous questions may be useful! (d) Try to find a way to represent each finite subset of and that will give you a unique number for each subset. Q9 – If we have n+1 terms and n possible values, then there must be a repeat somewhere…

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