0
votes
2answers
43 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
votes
4answers
30 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
votes
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 ...
15
votes
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
votes
2answers
45 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 ...
1
vote
2answers
64 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?
3
votes
1answer
94 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 ...
1
vote
8answers
125 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 ...
1
vote
4answers
81 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 ...
0
votes
2answers
64 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 ...
6
votes
3answers
327 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 ...
2
votes
3answers
203 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: ...
14
votes
2answers
338 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 ...
1
vote
2answers
135 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 ...
3
votes
4answers
222 views

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 ...
2
votes
1answer
54 views

Löb's theorem and working with connectives of two levels

I try to decipher Löb's theorem by getting rid of some of the material implications for something more intuitive and in using classical substitutions for connectives on both levels. I came up with the ...
-1
votes
3answers
233 views

Two easy proofs by contradiction

Check the validity of the statements below using contradiction method (i) p: The sum of an irrational number and a rational number is irrational (ii) q: If $n$ is a real number with $n ...
2
votes
5answers
417 views

Is my proof that $(p \wedge \neg p) \Rightarrow q$ correct?

I was asked by a professor a while ago to prove $(p \wedge \neg p)$ implies $q$. Whether through laziness or cleverness, I came up with the following proof: $p \wedge \neg p$ (by assumption). ...
2
votes
2answers
120 views

help me define the connectives for 3 value logic

so basically i have a project about 3 valued logic ie truth=1 false = 0, unknown = 1/2 in a previous project I had to come up with formulae for 2 valued logic as follows: ...
5
votes
4answers
376 views

Can the principle of explosion be removed from constructive logic?

Classical logic has the theorem ($p\wedge\lnot p)\rightarrow q$, which I will call EFQ ("ex falso quodlibet"). Constructive logic often has the principle built in, in the form of an axiom ...
9
votes
5answers
410 views

How do we know that we'll never prove a contradiction in Math

I know that we can prove a contradiction in naive set theory. Let D be a set of all sets that don't contain itself. Say D does not contain D. Then D contains D. That means D contains itself. A ...