Since there are two 0000001456 00000 n 0000043714 00000 n startxref $1,000,000 Question . 0000022431 00000 n Proof by Cases . Showing that the statement follows in all cases. 0000006280 00000 n February 11, 2013.
0000005568 00000 n (Proof by Cases) Albert R Meyer cases.9 115 0 obj <>stream 0000000016 00000 n 1.7 Proof by Cases. Is P = NP ? The answer is on my desk! 0000016857 00000 n $1,000,000 Question . 0000002086 00000 n Theorem 1. 0000005872 00000 n To prove a statement P is true, we begin by assuming P false and show that this leads to a contradiction; something that always false. 0000004174 00000 n Some philosophers* think reasoning this way is worrisome. %PDF-1.6 %���� 0000039595 00000 n Breaking a complicated proof into cases and proving each case separately is a com mon, useful proof strategy. 0000006544 00000 n %%EOF 0000040718 00000 n There are only two steps to a direct proof (the second step is, of course, the tricky part): 1. 0000042073 00000 n 1.1 Direct Proof (Proof by Construction) In a constructive proof one attempts to demonstrate P )Q directly. 1.1.3 Proof by cases Sometimes it’s hard to prove the whole theorem at once, so you split the proof into several cases, and prove the theorem separately for each case. 0000036806 00000 n
Proof by Contradiction This is an example of proof by contradiction. 0000019124 00000 n 0000042632 00000 n Use P to show that Q must be true. 0000015695 00000 n
0000035707 00000 n 0000020317 00000 n 16 . 0000003971 00000 n Example: Let n be an integer. 0 Dividing the situation into cases which exhaust all the possibilities; and 2. 0000002922 00000 n
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0000013672 00000 n cases.7 Albert R Meyer February 11, 2013. 0000041720 00000 n
It's important to cover all the possibilities. Reasoning by cases can break a complicated problem into easier subproblems. 0000022141 00000 n Proof by Cases (Example) •Proof (continued): 0000030516 00000 n Here’s an amusing example. 0000040951 00000 n 0000014084 00000 n This proof is carried out in very much the same way as the direct proof in Example 2.3.1. 0000043188 00000 n And don't confuse this with trying examples; an example is not a proof. 0000042713 00000 n
0000041030 00000 n 0000004675 00000 n You can sometimes prove a statement by: 1. Show that if n is not divisible by 3, then n2 = 3k + 1 for some integer k. Proof by Cases (Example) •Show that if an integer n is not divisible by 3, then n2 = 3k + 1 for some integer k. •Proof : n is not divisible by 3 is equivalent to n = 3m + 1 for some integer m or n = 3m + 2 for some integer m _. h�b```b``�������� Ȁ ��@Q��A�j. Now we give a direct proof of the contrapositive: we assume mand nare arbitrary odd integers and deduce mnis odd. NEGATION 3 We have seen that p and q are statements, where p has truth value T and q has truth value F. The possible truth values of a statement are often given in a table, called a truth table. 0000031692 00000 n 0000033610 00000 n trailer
0000042009 00000 n 0000004640 00000 n 0000003444 00000 n Assume that P is true. Many of the statements we prove have the form P )Q which, when negated, has the form P )˘Q. 0000013301 00000 n *intuitionists. 58 0 obj <> endobj 0000021938 00000 n Examples of Proof by Induction Introduction: “A Journey of a thousand miles begins with a single step” This phrase rather nicely sums up the core idea of proof by induction where we attempt to demonstrate that a property holds in an infinite, but countable, number of cases, by extrapolating from the first few. 0000017938 00000 n 0000004487 00000 n 0000002119 00000 n 0000015224 00000 n 0000043406 00000 n Let’s agree that given any two people, either they have met or not. 0000004901 00000 n 0000030141 00000 n 1.3. 58 58 0000039498 00000 n 0000005361 00000 n The truth values for two statements p and q are given in Figure 1.1. 0000004833 00000 n 0000001958 00000 n In this particular example, the necessity of De Morgan’s Law may be more evident in the symbolic form than in the \English version." 0000034657 00000 n If every pair of people in a group has met, we’ll call … 0000004752 00000 n 2. xref
0000021415 00000 n 0000001789 00000 n 0000002679 00000 n This is the simplest and easiest method of proof available to us. 0000044123 00000 n 0000042427 00000 n
Proof by Cases.
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