Leaving Cert Mathematics Proofs and Constructions: Explain, with the aid of an exam

Question

Explain, with the aid of an example, what is meant by proof by contradiction.
Note: you do not need to provide the full proof involved in your example. Give sufficient outline to illustrate how contradiction is used.

Explanation:

To prove a statement by contradiction, we assume that the statement is false and then prove that this assumption contradicts another statement that is known to be true.

Example:

Consider the statement
The lines $y = x$ and $y = x + 1$ do not intersect.

We can prove this by contradiction as follows.
Assume that the statement is false. So there is some point $(a, b)$ that lies on both lines. Therefore $b = a$ since the point is on the line $y = x$ and $b = a + 1$ since the point is on the line $y = x + 1$.
Combining these equation, we get 
$a = a + 1$ 
which implies that 
$0 = 1$.
However this contradicts that fact that $0 
eq 1$. Therefore our original assumption is false. So we have proved that the lines $y = x$ and $y = x + 1$ do not intersect.

Explain, with the aid of an example, what is meant by proof by contradiction - Leaving Cert Mathematics - Question 6A - 2014

Worked Solution & Example Answer:Explain, with the aid of an example, what is meant by proof by contradiction - Leaving Cert Mathematics - Question 6A - 2014

Explanation:

Example:

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