For questions about contradictions, and proof by contradiction.

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1answer
32 views

What means $A \subsetneq X$ with A ~ X? [on hold]

How it is possible to have a subset A, which is $\neq$ to X and at the same time they have an equivalence relation ~? When $A \subset X$ therefore a $\in$ A is also a $\in$ X. With A ~ X ...
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1answer
19 views

A question about proof by contradiction

$1$. Claim: $\varnothing$ is an antichain. $Proof$: Suppose $\varnothing$ is not an antichain. Then $\exists$ a pair $x, y \in \varnothing$ such that $x$ and $y$ are comparable. Contradiction: ...
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2answers
44 views

How do I prove this using proof by contradiction

There is a set a set $S$ of numbers. i.e. $(s_1, s_2, s_3, s_4, s_5, ..., s_n)$. The average of the numbers in the set is $N$. How do I prove that at least one of the numbers in the set is greater ...
2
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1answer
64 views

1.Why is this proof of “$\sqrt{2}$ is irrational” titled as “Proof by infinite descent”?2. Do I understand it correctly?

I am reading this wikipedea article on the proof of irrationality of $\sqrt{2}$. It uses the principle of infinite descent. I understand it as: We assume $\sqrt{2}=\dfrac pq$, where $p$ and $q$ are ...
5
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1answer
127 views

Understanding the proof of “$\sqrt{2}$ is irrational” by contradiction.

I have some difficulties in understanding the proof of "$\sqrt{2}$is irrational" by contradiction. I am reading it in 10th class(in India) Mathematics book( available online, here ) This is the ...
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4answers
87 views

For every $x \in [\frac{\pi}{2},\pi]$, $\sin(x)+\cos(x)\geq 1$. Prove rigorously by contradiction.

For every $x \in [o,\frac{\pi}{2}]$, $\sin(x)+\cos(x)\geq 1$. How do you prove this rigorously by contradiction? I understand you start by assuming that $\sin(x)+ \cos(x)<1$ and prove this is a ...
0
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2answers
18 views

Proof by contradiction how to show is properly

For every $x \in \left[\pi/2,\pi\right]\,,\ \sin\left(x\right) − \cos\left(x\right) \geq 1$. I have drawn the graph and can clearly see that A is true however how do I prove it correctly.
0
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1answer
65 views

prove pigeonhole problem using contradiction

I'm trying to prove this pigeonhole problem: Given that fact that $\lceil x \rceil < x + 1$, give a proof by contradiction that if $n$ items are placed in $m$ boxes then at least one box must ...
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2answers
76 views

prove by contradicition that there does not exist a constant $c$, such that for all integers $n \geq1$, $(n+1)^2 - n^2 < c$

First thing I did was simplify the above to $2n-1 < c$, from there I changed the simplified expression to $2n-1 \geq c$ and tried to make that contradict with $n \geq 1$. the problem being that I ...
4
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1answer
105 views

Proof by counter example of optimal solution for Coin Changing problem (no nickels)

I'm a tutoring a student whose working on the classical coin changing problem. For those who are unfamiliar with problem or the greedy algorithm used for it. The goal is find the fewest number coins ...
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2answers
100 views

Proof Question using Proof By Contradiction, irrationality of $a + \sqrt[b]{5}$

I answered this question but am not quite sure if i did what was correct. If anything is wrong please point it out, thanks. Question:Prove that any number of the form $a + \sqrt[b]5$ is irrational, ...
0
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1answer
35 views

Prove using a proof by contradiction: There is no smallest positive real number

Prove using a proof by contradiction: There is no smallest positive real number Let us assume the contraction: There is a smallest positive real number. How do I continue?
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3answers
99 views

Show that there are no positive integer solutions to x^2 + x + 1 = y^2.

I'm trying to prove that $x^2 + x + 1 = y^2$ has no integer solution, but I'm having a lot of trouble. So far I've tried proof by contradiction, but all of that seems to rely on me being able to ...
0
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2answers
45 views

Are there multiple contrapositives?

Are there "multiple contrapositives"? Normally a contrapositive from P implies Q changes to not Q implies not P. Secondly, can a contrapositive be in the from of P in the antecedent and Q be the ...
0
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4answers
42 views

Proof by contradiction involving set theory

Using a contradiction, prove the following: If $S\cap T = \emptyset$ and $S\cup T = T$, then $S = \emptyset$. So far, I've written the definitions of the intersection and union, and I've ...
4
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2answers
955 views

“A Proof by Contradiction”

Prove that for all $x,y \in \mathbb{Z}$, $x^2 - 4y \ne 2$. Using a contradictory method would be appropriate. So, for this question, I assume, for the sake of a contradiction, that There exists ...
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2answers
42 views

Do we simplify the Proof by Contradiction?

Prove the following by contradiction: Suppose $a,b\in\mathbb{Z}$. If $4|\left(a^2+b^2\right)$, then $a$ and $b$ are not both odd (in other words, $a$ and $b$ are even) So, I did this: Assume $a$ ...
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0answers
53 views

Proving that there always exists two opposite points on a circle where the temperature difference is less than 1

You are given $n$ ($n$ is even) integers $a_0,a_1,\ldots,a_{n-1}$ representing temperature measurements, equally spaced around a circle. Since the points are "close", the temperature difference ...
0
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4answers
32 views

Show that if n is an integer and 3n+ 2 is even, then n is even using contradiction

Show that if $n$ is an integer and $3n+ 2$ is even, then $n$ is even, using a proof by contradiction. That's the question. So since we're using contradiction, I need to show that N is odd and prove ...
5
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1answer
211 views

If $x\rightarrow y$ and $y \rightarrow z$, prove, by contradiction, that $x \rightarrow z$

Say you're given $$x\Rightarrow y$$ $$y\Rightarrow z$$ Prove that $x\Rightarrow z$ by contradiction. It seems like such a simple task, because it's easy to evaluate that it must be true. But I ...
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2answers
203 views

Prove that if all edge-costs are different, then there is only one cheapest tree.

Prove that if all edge-costs are different, then there is only one cheapest tree (minimum spanning tree). (Use contradiction and make sure to keep track of the costs of the different trees involved.) ...
0
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1answer
56 views

is this proof by contradiction fine? sum of composite numbers

I have seen the following problem: Any number n>11 can be formed by the sum of two composite numbers and it says to prove it by contradiction. I have done the following: Assume that the sum will be ...
15
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10answers
2k views

Having hard time understanding proofs by contradiction.

I am reading an introductory book on mathematical proofs and I don't seem to understand the mechanics of proof by contradiction. Consider the following example. $\textbf{Theorem:}$ If $P \rightarrow ...
2
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2answers
51 views

Proof by Contradiction with Multiple Axioms

Looking at proofs by contradiction and it seems I've run into something that does not sit well with me. I am fine with the law of the excluded middle (thus not an intuitionist) and more fundamentally ...
3
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4answers
58 views

Proof by contradiction that $(n+1)^3 \not= n^3 +(n-1)^3$ for $3$ consecutive positive integers

Prove by contradiction that if n-1, n, n+1 are consecutive positive integers, then the cube of the largest cannot be equal to the sum of the cubes of the other two. Assume that: $$ (n+1)^3 = ...
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1answer
64 views

Help prove $f:X \rightarrow Y$ is an injection $\Leftrightarrow$ $f:X\rightarrow Y$ is a surjection when $|X|=|Y|$

I need to prove: Given non-empty finite sets $X$ and $Y$ with $|X| = |Y|,$ a function $X\rightarrow Y$ is an injection if and only if it is a surjection. The hint given is to use the pigeonhole ...
0
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3answers
152 views

Proof by contradiction using counterexample

Why can't we use one counter example as the contradiction to the contradicting statement? Example: Let a statement be A where a-->b. We can prove A is not true by finding a counter example. Now, in ...
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0answers
42 views

four point in a row

We have painted all dots of page with two colors(blue and green), proof that there are four point with green color in a line that distance of any two neighbors of this four is one unit or there are ...
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2answers
109 views

Prove: If $a\in\mathbb Z$ and $|a| > 1$, then $1/a \notin \mathbb Z$.

Prove: If $a$ is an integer and $|a| > 1$, then $1/a$ is not an integer. Hi, I need help proving this either by contradiction or contrapositive. I'm not sure where to begin
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1answer
62 views

Check workings for Strong Induction (Proof by Contradiction)

I want to prove the following: Suppose that $P(n)$ is a statement involving a general positive integer $n$. Then $P(n)$ is true for all positive integers $n$ if: i) $P(1)$ is true, and ...
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2answers
41 views

Contrapositive proof using rule of divisibility

Suppose $x,y,z$ are integers and $x \neq 0 $ if $x$ does not divide $yz$ then $x$ does not divide $y$ and $x$ does not divide $z$. So far I have: Suppose it is false that $x$ does not divide $y$ and ...
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0answers
44 views

What is the relationship between division and proving an integer is odd?

I am trying to use proof by contradiction to prove: $101$ is an odd integer. I know that the first step is to assume that $101$ is even, so: $101 = 2q, q \in \mathbb{Z}$ Then I am stuck. I don't ...
2
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1answer
64 views

About an impossibility

Suppose Goldbach conjecture true: $\forall{n\geq{9}}$ integer, $\exists{p_1\geq{3},p_2\geq{3}}$ primes for which $2n=p_1+p_2$. But $\forall{m>1}$ integer, $\forall{p_1\geq{3},p_2\geq{3}}$ primes, ...
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5answers
54 views

How to prove that $x^2≡2(\bmod 3)$ is not a complete square

Let $m$ be the product of first n primes (n > 1) , in the following expression : $$m=2⋅3…p_n$$ I want to prove that $(m-1)$ is not a complete square. I found two ways that might prove this . My ...
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2answers
88 views

Query about Reductio Ad Absurdum

If we use the method of contradiction(i.e.Reductio Ad Absurdum), and if one of our assumptions is wrong, does that mean that all our assumptions are wrong and is the statement or hypothesis proved?
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1answer
60 views

Mathematical Proof (Apostol)

If $x > 0$, prove that there is a positive integer $n$ such that $\frac{1}{n} < x$ byy either contradiction or contrapositives. My attempts By contrapositives: Givens by contrapositive method ...
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0answers
76 views

Can proof by contradiction and counterexample by used at the same proof?

Here is a part of a theorem: If $\alpha>1$ and $x\ge-1$ then $(1+x)^\alpha \ge 1 + \alpha x$ I was wondering if I could use proof by contradiction and counterexample at the same time. Assume ...
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1answer
19 views

Struggling with proof, by contrapositive?

I am having trouble solving this proof. I tried to do a proof by contrapositive. Q = $(u+z)/(v+w) < z/w$ P = $(u/v < x/y \land x/y < z/w)$ Assuming $\lnot Q$ got me: $u/v \ge z/w$ ...
2
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1answer
83 views

Counter-example to contraction principle

One of the conditions of contraction principle is $d(fx,fy)<cd(x,y)$ for all $x,y$ where $c<1$. Now I am finding an example that when the condition is replaced by $d(fx,fy)<d(x,y)$, the ...
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10answers
1k views

Challenge: Demonstrate a Contradiction in Leibniz' differential notation

I want to know if the Leibniz differential notation actually leads to contradictions - I am starting to think it does not. And just to eliminate the most commonly showcased 'difficulty': For the ...
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0answers
135 views

Proof by contradiction problem on rational numbers

Using proofs by contradiction, show that there is no smallest negative rational number and no largest positive rational number. Assume that there is a smallest negative rational number. Therefore, ...
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4answers
69 views

Show that if $3\mid(a^2+1)$ then $3$ does not divide $(a+1)$.

Show that if $3\mid(a^2+1)$ then $3$ does not divide $(a+1)$. using proof of contradiction can someone prove this using contradiction method please
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3answers
57 views

proof by contradiction that if a and b are positive integars and $ab >100$ then at least one of the integars a and b is greater than 10 [closed]

does anyone know how to proof by contradiction that if $a$ and $b$ are positive integars and $ab >100$ then at least one of the integars $a$ and $b$ is greater than $10$
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2answers
151 views

Suppose F and G are families of sets. Prove that F and G are disjoint iff for all A∈F and B∈G, A and B are disjoint

I am trying to work through this homework problem but I am having trouble getting past how to get started. Could help with setting up this to prove? I know I need to prove by contradiction and ...
3
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1answer
95 views

Theorems that we can prove only by contradiction

While most of the world is fine with proofs performed by contradicting the thesis, direct proofs are sometimes considered more elegant than indirect ones. Those who prefer intuitionism or ...
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8answers
140 views

Prove if $n^3$ is odd, then $n^2 +1$ is even

I'm studying for finals and reviewing this question on my midterm. My question is stated above and I can't quite figure out the proof. On my midterm I used proof by contraposition by stating: If $n^2 ...
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1answer
192 views

[Beginner]How to tackle mathematical proofs?

So I recently joined university for a BSc in mathematics. I have never been exposed proofs but I have knowledge of algebra, trigonometry, and some differentiation/integration. Now I'm struggling with ...
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3answers
37 views

A proof using Fermat's Little Theorem?

Let $p$ be prime and let $a\in Z$ such that p doesn't divide a (sorry I couldn't find the symbol for it in MathJaX). Prove that if $k$ is the smallest integer such that $a^k\equiv 1 \pmod p$, then ...
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4answers
84 views

logic: two simple math contradictions

1.The contradiction of the sentence: - There is a greater number than a million. can be stated as follows: - There is a number which is not greater than a million. 2.and the contradiction of the ...
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2answers
47 views

Determining if $0$ is part of this set

Let $S$ be a set of rational numbers that is closed under addition and multiplication, and having the property that for every rational number $r$ exactly one the following three statements is true: ...