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My textbook has asked me to find the "contradiction" of a given statement but I have not learned to do such a thing and googling has not yielded in any results whatsoever. Exactly what is the contradiction of a statement and how to form it, given any statement?

Edit: This is the problem in question:

"Write the converse, contrapositive and contradiction of the statement "If ∆ABC is right angled at B, then AB²+BC²=AC²"

The answer for the contradiction of this statement is given as

"∆ABC is right angled at B and AB²+BC²≠AC²"

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    Never heard of that.. maybe it meant negation? or converse, in case of an implication? – Angelo Rendina Sep 15 '16 at 12:23
  • Are you sure that it's not contrapositive? What textbook are you using? Can you upload a relevant paragraph? – The Chaz 2.0 Sep 15 '16 at 12:41
  • Yes, I am absolutely sure that they mean neither the converse, nor the contrapositive. –  Sep 15 '16 at 13:05
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    From the answer, it is clear that the "contradictory" of $A$ is $\lnot A$. see example: the statement is : "If $p$, then $q$" and its contradiction (I would prefer: contradictory) is : "$p$ and not $q$". – Mauro ALLEGRANZA Sep 15 '16 at 13:14
  • I think (with @AngeloRendina ) that "contradiction" here means "negation". Can you find places in your textbook where the author uses either term? I've upvoted the question - clearing up textbook vocabulary is important. – Ethan Bolker Sep 15 '16 at 13:14
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    See contradictory. – Mauro ALLEGRANZA Sep 15 '16 at 13:15
  • Actually, my textbook does use the term "negation" quite distinguishably. However, I think it's using this term with "contradiction" interchangeably. Now I need to make sure that I have got it right by looking up whether this is the case by solving some more problems. I will post again, if I find that this is indeed the case. Thanks :-) –  Sep 15 '16 at 13:18

3 Answers3

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What you seem to be asking, when you (your text) refers to the "contradiction" of a statement, might be better termed as the "negation" of a sentence. For example, if person A declares $p$, another person B would be contradicting A by declaring "not $p$. That is, person B is asserting the negation of $p$.

So, we have a sentence we'll call $p$. It's negation is simply $"\lnot p".$ So if we know that $p$ is true, its negation would be false.

If the sentence (proposition), for example, is "$x$ is an odd number", it's negation would be "it is not the case that $x$ is an odd number," or equivalently, its negation would be "$x$ is an even number."

amWhy
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a contradiction is a statement which is always false, like P and notP.

for example if we have a<1 and we find that a>1, there is a contradiction.

a statement whis is always true is called a tautology ( tautologie in french). for example : the square of a real is positive.

hamam_Abdallah
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Let's use formal approach.

Let $$A = \text{∆ABC is right angled at B},$$ $$B = \text{AB²+BC²=AC²}.$$

Then your first statement is $A \to B$ that is equal to $\neg A \vee B$. The contradiction (using de Morgan's theorem) is $\neg(\neg A \vee B) = A \& \neg B$. And this stands for exactly '∆ABC is right angled at B and AB²+BC²≠AC²'.

Denis Korzhenkov
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  • Yes, I see now that the terms "contradiction" and "negation" can be used interchangeably. Thanks :-) –  Sep 15 '16 at 13:19