There are four basic proof techniques to prove p =)q, where p is the hypothesis (or set of hypotheses) and q is the result. 1.Direct proof 2.Contrapositive 3.Contradiction 4.Mathematical Induction What follows are some simple examples of proofs. You very likely saw these in MA395: Discrete Methods. 1 Direct Proof
EXAMPLE 1 Give a direct proof of the theorem “If n is an odd integer, then is odd.” Solution: n = 2k + 1 , : is odd ----- EXAMPLE 2 Give a direct proof that if m and n are both perfect squares, then nm is also a perfect square. (An integer a is a perfect square if there is an integer b such that a = .)
Primenumbers Definitions A natural number n isprimeiff n > 1 and for all natural numbersrands,ifn= rs,theneitherrorsequalsn; Formally,foreachnaturalnumbernwithn>1 ...
examples of how to write a proof correctly. Mathematical statements may be de nitions, or logical statements, and can express a complicated idea in a few words or symbols, as the following examples show. Thus until one gets used to the language it really can take a mental e ort to understand a mathematical statement.
For example, 1 + p xworks. On the other hand, when x<0, all squares are bigger than x, so fy: y2 >xg= (1 ;1) = R: So any value of yworks. We could take y= 0, for example. From all this preliminary analysis, one can extract the following proof. Proof. Given x, we need to produce ysuch that y2 >x. We break into cases according to whether x 0 or x<0.
For example, in the proofs in Examples 1 and 2, we introduced variables and speci ed that these variables represented integers. We will add to these tips as we continue these notes. One more quick note about the method of direct proof. We have phrased this method as a chain of implications p)r 1, r 1)r 2, :::, r
Common Proof Techniques (with examples) Linear Algebra MATH 228 Fall 2016 DefinitionVerification The simplest kind of proof. It generally comes in the form “Show that an X is a Y,” where X and Y are givenmathematicalobjects. Tocompletetheseproofs,yourefertothedefinitionofY anddemonstratethat thedefiningcriteriaforY aresatisfied. Theorem1.
symbols (such as equations, numbers and formulas). This will happen in most mathematical proofs. A mathematical argument should be made of complete sentences, containing words, symbols, or a combination of both. The words are meant to help the reader follow the logical ow of the argument, and to connect the various statements that appear in the ...
You should use these proofs as examples of how you should write a proof. In particular, notice that a proof consists of clear, short, precise, steps, each one of which makes a de nite assertion; and that at each point in the proof, it is clear (a) what we are assuming, (b) what we are trying to proof, (c) what are the values of all the letter ...
Introduction to mathematical arguments (background handout for courses requiring proofs) by Michael Hutchings A mathematical proof is an argument which convinces other people that something is true. Math isn’t a court of law, so a “preponderance of the evidence” or “beyond any reasonable doubt” isn’t good enough. In principle
2.9. Constructive proof. A constructive proof demonstrates the existence of a mathematical object by constructing it explicitly and showing that it has the required properties. More explicitly, this is a proof of a statement of the form pDx PAqpPpxqq. Such a proof involves constructing an element x in a set A and showing that it satisfies ...
Mathematical induction is a method that allows us to prove in nitely many similar statements in a systematic way, by organizing them all in a de nite order and showing ... EXAMPLES OF PROOFS BY INDUCTION 3 The proved case of positive exponents tells us (MA 1M 1)K = M(A 1)KM 1 by replacing Awith A 1. Feeding that into (2.1), (2.2) (MAM 1)k = M(A ...
Proof-writing examples Math 272, Fall 2019 Proof of Corollary 5 Suppose that A~v = ~0. Proposition 4 says that if A is invertible, then ~v = ~0. By the contrapositive, if ~v 6=~0, then A is not invertible, as desired. 4 Equality of sets It is frequently convenient to express certain if and only if statements as equation of sets. For
Example: If Ais the event that x 10, then :Ais the event that x>10. It is common to use mathematical symbols for words while writing proofs in order to write faster. The following are commonly used symbols: 8orF all, for any 9There exists 2Is contained in, is an element of 3Such that, contains as an element ˆIs a subset of
the methods of proofs. A number of examples will be given, which should be a good resource for further study and an extra exercise in constructing your own arguments. We will start with introducing the mathematical language and symbols before moving onto the serious matter of writing the mathematical proofs.
Proof: Assume 0 < c < d. If, on the contrary, √ c ≥ √ d, then the theorem above implies that √ c2 ≥ √ d 2, so c ≥ d. By assumption, this cannot be the case, so √ c < √ d. The proof of this corollary illustrates an important technique called ‘proof by contradiction’. The idea is to assume the hypothesis, then assume the ...
Examples of Mathematical Proofs Example 1 (Direct proof). Prove that the square of an odd integer is odd. Proof. Let n be an odd integer. Then n = 2m+1 for some integer m. Therefore n2 = (2m+1)2 = 4m2+4m+1 = 2k + 1 where k = 2m2 + 2m is an integer. Hence, n2 is odd. Example 2 (Proof by construction). Prove that the following statement is false:
Direct Proof Examples 1. Conjecture: n2 −3 is even if nis odd, n∈ Z. Discussion: The first thing to do is identify the hypothesis and the conclusion. Why? Because in a direct proof, we are allowed to assume the hypothesis, giving us a piece of information that we can use as a starting point for our argument.
Math 150s Proof and Mathematical Reasoning Jenny Wilson Common Mistake # 7. Imprecise statements. An argument cannot be rigorous if it involves ambiguous or poorly defined statements. Be specific. Use mathematical terminology, and use it correctly. For example, Imprecise Sentence: “The sine function looks the same every 2ˇ.” Better ...
