1.Direct proof 2.Contrapositive 3.Contradiction 4.Mathematical Induction What follows are some simple examples of proofs.
The L-notation of Table 3 will be used to nd the solution y(t) = 1+5t t2. The statement P1 says that x1 = 1 < 4, which is true. The proof by contrapositive is based on the fact that an implication is equivalent to its contrapositive. .
Proof. Example 2 Use the comparison test to determine if the following series converges or diverges: X1 n=1 21=n n I First we check that a n >0 { true since 2 1=n n >0 for n 1. IfP (A )µP B,then A µB.
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statement q is true.
2.6 Indirect Proof. 7 10.2 Equivalence class of a relation 94 10.3 Examples 95 10.4 Partitions 97 10.5 Digraph of an equivalence relation 97 10.6 Matrix representation of an equivalence relation 97 10.7 Exercises 99 11 Functions and Their Properties 101 11.1 Definition of function 102 11.2 Functions with discrete domain and codomain 102 11.2.1 Representions by 0-1 matrix or bipartite graph 103
n = 2k for some integer k. Multiply both sides by −1 .
Given: M is the midpoint of Prove: ≅ 9.
Solution: To gure out the matrix for a linear transformation from Rn, we nd the matrix A whose rst column is T(~e 1), whose second column is T(~e 2) { in general, whose ith column is T(~e i). Proof: Suppose n is any [particular but arbitrarily chosen] even integer.
. Solution e) None of the above.
I Using the ratio test.
Example 2.3.1 shows a simple direct proof of a very familiar result. Proof: By contradiction; assume √2is rational. [21 pts - 3 each] Prove each of the following claims by your choice of direct proof or proof by contra-positive.
Existence of Solution.
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(Direct .
I Since P 1 n=1 1 is a p-series with p = 1 (a.k.a.
Thus cont'd
Prepared by Professor Zoran .
Proof by Contrapositive July 12, 2012 So far we've practiced some di erent techniques for writing proofs.
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¥Use logical reasoning to deduce other facts.
identify the hypoth-esis and conclusion of the statement and the domain.
Proof 1 Formally the statement can be written as ∀x ∈ ¢ p → q where p and q are defined as "x 2 is even" and "x is even" respectively.
Proofs Basic proof methods: • Direct, Indirect, Contradict ion, By Cases, Equivalences Proof of quantified statements: • There exists x with some property P(x).
1.1 Direct Proof (Proof by Construction) In a constructive proof one attempts to demonstrate P )Q directly. [21 pts - 3 each] Prove each of the following claims by your choice of direct proof or proof by contra-positive. 3.3: Indirect Proofs.
90 DirectProof Definition4.4 Suppose aandb areintegers.
Spans Last time, we saw a number of examples of subspaces and a useful theorem to check when an arbitrary subset of a vector space is a subspace. Proof Templates Page 1 Example Direct Proof If A, then B If x + 10 is odd, then x is odd Proof Proof 1.
A simple direct proof example on the sum of the first n natural numbers (starting from 1).
Example: Give a direct proof of the theorem "If n is an odd integer, then n^2 is odd." Solution: Assume that n is odd. In . Supposethenumbersa andb areeven. . Quite frequently you will find that it is difficult (or impossible) to prove something directly, but easier (at least possible) to prove it indirectly. Theorem Let {a n} be a positive sequence with lim n→∞ a n+1 a n = ρ exists.
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In order to illustrate this type of proof we assume that we know: 1.
For example, 5divides 15because ˘ ¢3.We write this as j. However, if we allow complex numbers, then both i and i are solutions to the .
These statements are considered facts, therefore, true. From this contradiction, then, we can only conclude that there is no integer solution to the equation
Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. Finding the coefficients, F' m, in a Fourier Sine Series Fourier Sine Series: To find F m, multiply each side by sin(m't), where m' is another integer, and integrate: But: So: Åonly the m' = m term contributes Dropping the ' from the m: Åyields the coefficients for any f(t)! There are two hypotheses, \mis an odd
10.5) I The ratio test. This example was created in a dynamic mathematics software program which gave these related measurements. 1 Example (Laplace method) Solve by Laplace's method the initial value problem y0 = 5 2t, y(0) = 1.
What is a proof?
Dropping the inessential factor of 2, we conclude that a real solution to the two-dimensional Laplace equation can be written as the real part of a complex function.
Solution: Suppose that √ p is a rational number. Hildebrand Even/odd proofs: Practice problems Solutions The problems below illustrate the various proof techniques: direct proof, proof by contraposition, proof by cases, and proof by contradiction (see the separate handout on proof techniques). Abel, at the age of 19, gave a complete proof of the non-existence of the quintic formula in 1821.
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AssumeP(A)µP(B).
(x, y are integers.)
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View Answer Under the law of large number, the variation of a statistic such as sample mean: a. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function f ∘ g in terms of the derivatives of f and g.
. Use a direct proof to prove that the square of an even number is an even number.
Result 4.2.
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. By our earlier result, since p2 is . 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.
But if the answer to a question is a proof, rather than a number or an expression, then the reader The essence of the idea is simple: for example, suppose you want to know whether it is overcast or sunny, but you can't see the sky through your window.
. If xz0 and yz0, then x yz 0 c. The product of two non -zero real numbers is a no n -zero real number.
the harmonic series), it
Proof by Contradiction This is an example of proof by contradiction.
(a) If x and y are both odd, then x+y is even. Proofs 4. (b) If ρ > 1, the series [We must show that −n is even.]
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Logic 14 Direct Proofs and Counterexamples II - Rationals SOLUTIONS 7. a. x,y R, If x y 0, then x 0 or y 0 b. Logic 2. 136 ProofsInvolvingSets Example8.9 Suppose A andB aresets.
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Start by assuming the Proof By Cases — Example.
Examples of Direct Method of Proof .
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Notice that we began with our assumption of the hypothesis and our definition of odd integers.
(a) If x and y are both odd, then x+y is even.
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• For all x some property P(x) holds. . a) The theorem is false but the proof is correct. Thus x2 + 1 < 0 is false for all x ∈ S, and so the implication is true.
Consider the sequence of real numbers de ned by the relations x1 = 1 and xn+1 = p 1+2xn for n 1: Use the Principle of Mathematical Induction to show that xn < 4 for all n 1. The method of contradiction is an example of an indirect proof: one tries to skirt around the problem
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b) The proof contains arithmetic mistakes which make it incorrect. Example 1 (Version I): Prove the following universal statement: The negative of any even integer is even.
Theorem A. Similarly 8j 32because ˘ ¢4,and¡ 6j because 6˘¡ ¢¡1.However, 6 does not divide 9 because there is no integer c for which 9˘ 6¢c.We . Proof.
. Linear Algebra Igor Yanovsky, 2005 7 1.6 Linear Maps and Subspaces L: V ! 10. 2.1 Set Theory A set is a collection of distinct .
Butthen,sinceP (A )µP B,itfollowsthat a
Base Case. Direct Proof: Example Theorem: 1 + 2 +h3 +rÉ + n =e n(n+1 . If f(z) is a complex function, then its real part u(x,y) = Re f(x+ iy) (2.6)
f (t) = 1 π F m′ sin(mt) m=0 ∑∞ 0
Here's a big hint: whenever we have absolute value - use proof by cases! This property is used to simplify the graphical convolution procedure.
State what to prove. ker(L) is a subspace of V and im(L) is a subspace of W.Proof. . Example 0.0.4.Measure of Arc In this example we cut our a segment of great circle by relating an inner angle. Proof by contradiction relies on the simple fact that if the given theorem P is true, then :P is false.
. to reach the result. . There are certain rules that must be followed
The proofs of Properties 3) and 6) are omitted. .88
Use rules of inference, axioms, and logical equivalences to show that q must also be true. The general format to prove \(P \imp Q\) is this: Assume \(P\text{. b a r for some integers a and b with b .
First we show there is always a solution. Proof: 9.
For example E ˘ ' 2 n: 2 Z " ˘ ' n : n isaneveninteger " ˘ ' n : n ˘ 2k,k 2 Z ".
In standard introductory classes in algebra, trigonometry, and calculus there is currently very lit-tle emphasis on the discipline of proof. Example #1 - Valid Claim. Writing Direct Proofs 1. Thus, n 2 is odd. 3.2 Direct Proofs Direct Proof of P ⇒ Q: Assume that P(x) is true for an arbitrary x ∈ S, and show that Q(x) is true for this x. .
Relations and Functions .
We are using the familiar de nitions of what it means for an integer to be even or odd: An integer nis even if n= 2kfor some integer k; an integer nis odd if n= 2k+ 1 for some integer k. Study the form of this proof. . Assume that P is true. Inductive Step.
There are two kinds of indirect proofs: the proof by contrapositive, and the proof by contradiction.
d) The proof is a correct proof of the stated result. Negating the two propositions, the statement we want to prove has the form Weusedirectproof.
For
The P s are the hypotheses of the theorem.
Through a judicious selection of examples and techniques, students are presented
Proof Sum Two Odd Integers Even. . Learn the process of indirect proofs through this free math video tutorial example of an indirect proof in geometry by Mario's Math Tutoring.0:12 Example 1 G.
Wesaythat dividesb, written aj b,if ˘ac forsome c2Z.Inthiscasewealsosaythat isa divisorof b,andthat isamultipleofa. . .
For this example let R= 4:69 and = :93759 radians, thus (4:69)(:93759) = 4:39, which is the measure of the length of the arc.
There are only two steps to a direct proof (the second step is, of course, the tricky part): 1.
1 Direct Proof Direct proofs use the hypothesis (or hypotheses), de nitions, and/or previously proven results (theorems, etc.)
Based on the assumption that P is not true, conclude something impossible.
Just as the proof above for 2, this shows that p divides n which means that p2 divides n2.
Often proof by contradiction has the form . . . Proofs can come in many di erent forms, but mathematicians writing proofs often strive for conciseness and clarity...well, at least they should be clear to other mathematicians.
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(a) If ρ < 1, the series P a n converges. I Comment: The root test. 3.
By definition of even number, we have. ! This implies that there is some integer k such that n = 2k + 1.
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Conclude, therefore, that P is true. Math 235 - Dr. Miller - HW #6: Direct and Indirect Proofs - SOLUTIONS 1.
There is one particularly useful way of building examples of subspaces, which we have seen before in the context of systems of linear equations.
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W is a linear map over F. The kernel or nullspace of L is ker(L) = N(L) = fx 2 V: L(x) = 0gThe image or range of L is im(L) = R(L) = L(V) = fL(x) 2 W: x 2 Vg Lemma. The following are the most important types of "givens.''.
- Proofs of 'For all x some property P(x) holds' must cover all
Clearly state all assumptions, and use rigorous logic (i.e., don't skip steps). actually only superficial, and the two proof techniques are equivalent.
Thus: x = a + b + c = 2i + 2j + 2k = 2(i + j + k) Since i + j + k is an integer, then x is even, a contradiction.
Hypotheses : Usually the theorem we are trying to prove is of the form. An important observation is the absolute value is a function that performs different operations based on two cases x < 0 or x ≥ 0. Example 2 - Solution Proof: Suppose r and s are rational numbers.
Then there exist two integers, n and m with no common divisor such that √ p = n/m. Example Directly prove that if n is an odd integer then n2 is also an odd integer. •Using either vertical or horizontal strips, perform a single integration to find the first moments. .
. P 1 ∧ … ∧ P n ⇒ Q.
Thus, in any problem in which you are asked to provide a proof, your solution will not simply be a short answer that you circle.
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These problems have been collected from a variety of sources (including the authors themselves), including a few problems from some of the texts cited in the references.
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Ratio test (Sect.
¥Keep going until we reach our goal.
For example, if there was a problem asking you to evaluate lim x!1 x4 1 x 1; it would not be acceptable to just write down \4." This would be unacceptable because there's no way for the person reading your answer to see why the limit should be 4. Direct Proof (Example 1) Show that if n is an odd integer, then n 2 is odd.
Direct proofs are especially useful when proving implications.
The proof began with the assumption that P was false, that is that ∼P was true, and from this we deduced C∧∼. A more direct proof of the following key result will appear in Theorem 4.1 below.
Instead of proving directly, it is sometimes easier to prove it indirectly.
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