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Information about Proof writing tips

proof writing tips for Math 189, BSU, spring 2014

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So x ∈ C. So A ⊆ C. Pretty straight forward. Now let’s try contrapositive. Proof. By contrpositive. First consider what the contrapositive of the statement is. We have ¬Q → ¬P. So we have A C → A B or B C. (Ask yourself why this is now an ”or” statement) Since A C, then ∃x ∈ A such that x ∈ C. / Choose this x. Now consider the relationship x has with B. Either x ∈ B or x ∈ B c . Case 1: Suppose x ∈ B. Then A ∈ B. However, since x ∈ C, then B C. / c . Then x ∈ A and x ∈ B. So A Case 2: Suppose x ∈ B / B. Since either case satisﬁes one of the things we need, then we are done. So, If A ⊆ B and B ⊆ C, then A ⊆ C by contrapositive. Now let’s consider this problem by contradiction. Proof. Suppose that if A ⊆ B and B ⊆ C, then A C. Assume A ⊆ B and B ⊆ C is true. If A C, then existsx | x ∈ A and x ∈ C. / Choose this x. Since A ⊆ B, then x ∈ B. Since B ⊆ C, then x ∈ C. However, we chose the x that is not in C. So we have a contradiction. So, if A ⊆ B and B ⊆ C, then A ⊆ C. So you can see how these proofs can be done by any of these methods. 2 Use deﬁnitions Often times problems can be unclear what is being asked right away. The best way to analyze what is being asked is to use deﬁnitions. For example, if you were given a problem that says ”Prove that (A∩((B ⊕C)A))∪A = A. First, you need to unpack what you have. Like we have always done, use the order of operations. Start inside the smallest parenthesis. 2

B ⊕ C = B ∪ C − B ∩ C. This means that we have everything in B and everything in C, but not the things in both. (So we have unpacked what the symbols means and we can understand what is going on.) So we can put this substitute this into our original statement. (A ∩ ((B ⊕ C)A)) ∪ A = (A ∩ ((B ∪ C − B ∩ C)A)) ∪ A. Now we can unpack the next thing. (B ∪ C − B ∩ C)A) means (B ∪ C − B ∩ C) ∩ Ac ). So we have everything in B and everything in C, but not the things in both intersected with all the things not in A. So if we were to describe the elements we are dealing with, we would say: {x | x ∈ A / and (x ∈ B or x ∈ C, but x ∈ B ∩ C.)} / Now if we were to put this in conjunction with our next requirement, we would have A ∩ ((B ∪ C − B ∩ C) ∩ Ac ). So now we can see that we have the intersection of all the things in A with a set containing only things not in A. So we don’t have any elements that are in A and not in A. So we have the empty set ∅. So (A ∩ ((B ⊕ C)A)) = ∅. So we can substitute this equivalence into the original equation and we get (∅) ∪ A), which equals A. This is what we sought out to prove. Note that we did this all without having to deal with a speciﬁc element and didn’t need to chase it around. It all came from unpacking deﬁnitions. It is a very handy and vital tool to have. 3 Put new ideas on new lines and be clear that you are onto a new thought. This may seem overly simple, but it makes everything so much easier. Proofs are easier to grade, logic is easy to follow, and mistakes can be easily identiﬁed. Let me show you what I mean. Consider the following proofs for the problem ”If A ⊆ B and B ⊆ C, then A ⊂ C.” Proof (incorrect): Proof (correct): Assume A ⊆ B and B ⊆ C. Assume A ⊆ B and B ⊆ C. Let x ∈ A, x ∈ B, x ∈ C. Let x ∈ A. So A ⊆ C. Since A ⊂ B by assumption, x ∈ B. Since B ⊆ C by assumption, x ∈ C. So A ⊆ C. 3

4 USE MORE PAPER This is the best advice I can give to any student of mathematics. The paper is dead. Use it. You can recycle stuﬀ later. For now, use the paper. This goes in hand with the last section. Take your time, use the space. There are a number of reasons for this. First, when your papers are being graded, it makes it a lot easier to provide feedback if you leave a visible space between problems, put ideas on new lines, and take up space. If you are trying to cram your entire proof onto two lines, there’s no room for Jason to add comments to your work. Second, if you take up space, it’s much easier for you to follow your reasoning as you look back over your problems if you get stuck. If you are cramming everything onto a couple of lines, you’re going to ﬁnd it much more diﬃcult to follow your own reasoning later, especially if you’re using your work to study from and it’s got some comments from Jason crammed in there somehow, making one big cluttered mess of work. Finally, if you aren’t worried about taking up space, then that’s one less thing you’re worried about. That’s always a win. 4

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