# Tagged Questions

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### Prove the following using the Real Theorem [closed]

I came across this question in a book and had difficulties in solving it: Let x,y belong to R and a>0 then show that ...
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### 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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### 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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### 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 ...
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### 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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### How is this a proof of the irrationality of $\sqrt2$

Proof. Suppose for the sake of contradiction that $\sqrt2$ is rational, and choose the least integer, $q \gt 0$, such that $(\sqrt2 − 1)q$ is a non negative integer. Let $q':=(\sqrt2 − 1)q$. Clearly ...
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### Prove that f is identically 0

I am trying to prove this but it seems to me something went wrong. could you help me to prove it by contradiction? Let $(x_m)$ be a real sequence. Let $f:\Bbb{R}\to\Bbb{R}$ be a function such that ...
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### If $2$ divides $p^2$, how does it imply $2$ divides $p$?

I'm trying to understand a proof by contradiction. It's proving by contradiction that $\sqrt2$ isn't rational. (It's a standard proof involving $\sqrt2=\frac{p}{q}$, where $p,q$ are already ...
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### continuous and bounded function without maximum or minimum

Give an example that contradicts this sentence : $f:(0,1]\to\Bbb R$ is a continuous and bounded function in $(0,1]$ then : $f$ has maximum or minimum. I have understood that $\sin(1/x)$ could be a ...
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### Prove or disprove that every Boolean function can be expressed by using only the operator ↓

I know that the ↓ operator means "nor" but how do I prove/disprove that every Boolean function can be expressed using only this operator ? Induction ? Contradiction ? I have to idea where to begin. ...
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### Is this a correct proof by contradiction?

Prove that if a set $A$ of natural numbers contains $n_0$ and contains $k+1$ whenever it contains $k$, then $A$ contains all natural numbers $\geq n_0$. I have attempted a proof by contradiction as ...
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### prove that one of the digits 1,2…9 occurs infinitely often in the decimal expansion of pi

prove that one of the digits 1,2...9 occurs infinitely often in the decimal expansion of pi. you may use without proof the fact that pi is irrational. It is recommended using proof by contradiction. ...
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Suppose if it is rational: $9 ^ {\frac {1} {8}} = {\frac {m} {n}}$ I know what to do with relative primes. M and N are the relative primes. $(n \times 9 ^ {\frac {1} {8}}) = m$ $(n \times 9 ^ ... 3answers 64 views ### Prove that if$n^{2} - \left(n - 2\right)^{2}$is not divisible by$8$then$n$is even Let n be an integer. Prove that if$n^{2} -\left(n - 2\right)^{2}$is not divisible by$8$then$n$is even. Can anyone help me step by step to understand this. 1answer 70 views ### Proofs by contradiction and set theory I'm having trouble understanding proofs by contradiction. I'm running things by memory and not by understanding what a contradiction is. I'd like to know what we're assuming and how to start. My ... 3answers 62 views ### Is it right to say that: if$2a+1=2b$we have a contradiction? I am trying to prove by contradiction and I have reached the conclusion that$2a+1=2b$. Now I am tempted to say it's a contradiction and call it a night. Is it a contradiction? because one is even and ... 2answers 51 views ### Using induction to prove$a_n >2^n$For the sequence$a_n=2a_{n-1}+1$where$a_0=1$Show that$a_n>2^n$using induction. Use proof by contradiction (minimum counterexample). Attempt: 1. I assume, that ... 2answers 53 views ### Question Regarding Logical Contradiction Let's say I attempted to solve a logical statement in the form using contradiction:$\forall x \in \Bbb R, (P \implies Q)$Negated:$\exists x \in \Bbb R, (P \land \lnot$Q). Initially I did not ... 4answers 66 views ### Prove that if$3|(a^2+b^2)$, then$3|a$and$3|b$, where$a, b$are integers [duplicate] I would like to know how to prove the above statement by contradition. Somebody said that one should prove it by this method but I have no idea what it is. 1answer 51 views ### Is this proof correct? (GCD) If this proof is incorrect can someone tell me what is wrong with it, and which step is incorrect. Let a, b ∈ℤ If gcd(a, b) = 35, then 25 ∤ a or 25 ∤ b. Proof Consider the contrapositive: if 25|a ... 1answer 22 views ### proofs involving power sets and universal quantifiers Im having trouble solving with a proof problem "A is not equal to the Null class then the intersection of class A is a set" and help on proofing this? 3answers 289 views ### Does a proof by contradiction always exist? Good day, Usually, proofs by contradictions are the easier, and sometimes, even the only ones available. However, there are cases where the easiest proof is not the proof by contradiction. For ... 3answers 315 views ### Proof by induction or contradiction? I have to prove that$(4k + 3) ^2 - (4k + 3)$is not divisible by$4$. What would be the best approach for this, proof by induction or contradiction? I've tried both and haven't got very far. Any ... 1answer 68 views ### Prove the limit..$\varinjlim \sqrt{n^2+1}-n=0$. I need to prove that this converges to 0. Usung the definition of a sequence helps for the normal problems but for this I believe the triangle inequality is used at ... 3answers 146 views ### How to solve a statement with contradiction evidence? I'm trying to solve the statement below with contradiction evidence. If$(P \rightarrow Q)$and$(Q \rightarrow R)$is true, then$(P \rightarrow R)$is true. This is what i've done so far: ... 2answers 248 views ### What practical proofs work in intuitionistic but not minimal logic? Intuitionistic logic contains the rule$\bot \rightarrow \phi$for every$\phi$. In the formulations I have seen this is a separate axiom, and the logic without this axiom(?) is termed "minimal ... 1answer 106 views ### Is proof by contradiction always a sufficient proof technique? Is proof by contradiction always a sufficient proof technique ? A proof by contradiction has the form: Let$P$and$Q$be statements. If$ P \rightarrow Q \land \lnot Q $then you can conclude ... 3answers 123 views ### Real analysis …show if its a rational number Show$(3+(2\sqrt{2})^\frac{2}{3}$is not a rational number... My result$x^3-6x^\frac{3}{2}+7=0${-1,1,-7,7} will not equal to zero .. Is my polynomial acceptable? I used contradiction for this ... 2answers 510 views ### Contradict the Contraction Mapping Theorem I am trying to show that the function$f(x) = 2\pi+x-\tan^{-1}x$is contractive but has no fixed points. Finally I wish to conclude that it does not contradict the contraction mapping theorem.$f$is ... 3answers 323 views ### The contradiction method used to prove that the square root of a prime is irrational The contradiction method given in certain books to prove that sqare root of a prime is irrational also shows that sqare root of$4$is irrational, so how is it acceptable? e.g. Suppose$\sqrt{4}$is ... 2answers 109 views ### Proof by contradiction with two assumptions I'm curious whether the following technique has ever been used in a proof of something. Assume two propositions$A$and$B$, then derive a contradiction. Thus you know that either$\lnot A$or$\lnot ...
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The symmetric difference of two sets $A$ and $B$ is the set $A \vartriangle B = (A \setminus B) \cup (B\setminus A) = (A \cup B) \setminus (A \cap B)$. Prove that if $A \vartriangle B \subseteq A$ ...
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### What's $P$ and what's $Q$ in this classic proof of the irrationality of $\sqrt 2$?

In this proof extracted from the Wikipedia A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational. If it were rational, it could be expressed as ...
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### What's the difference between Complex infinity and undefined?

Can somebody please expand upon the specific meaning of these two similar mathematical ideas and provide usage examples of each one? Thank you!
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### 3D Geometry Proof by Contradiction /Contrapositive (high school)

Could someone evaluate my work? A plane perpendicular to one of 2 parallel lines is perpendicular to the other line also. My two column proof so far: Let AB || CD and AB be perpendicular to plane ...
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### Proving a function is big O

How would I go about proving a function is big O? Do I use the regular proofs (direct, contrapositive, contradiction)? Example: Prove that $x^n$ is $O(n!)$ for every real number $x$. My proof by ...