Appendix 1: Proofs in Mathematics | A1.6

Question 6

Prove by contradiction that two distinct lines in a plane cannot intersect in more than one point.

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Solution
Understand the Question
  • Proof by Contradiction: We assume the opposite (negation) of the statement we want to prove and show that this leads to a contradiction with a known mathematical axiom.
  • Fundamental Axiom: Exactly one unique straight line passes through any two distinct points in a plane.

Step 1 · Assume the Contrary

Let ll and mm be two distinct lines in a plane.

Assume on the contrary that ll and mm intersect at more than one point. Let them intersect at two distinct points, AA and BB.Diagram 1

Line l passes through A and BLine m passes through A and B\begin{aligned} &\text{Line } l \text{ passes through } A \text{ and } B \\ &\text{Line } m \text{ passes through } A \text{ and } B \end{aligned}

Step 2 · Arrive at a Contradiction

According to the fundamental axiom of geometry, exactly one unique line passes through two distinct points.

Line l passes through A and BLine m passes through A and BA and B are distinct pointsTherefore, line l and line m must be the same line\begin{aligned} &\text{Line } l \text{ passes through } A \text{ and } B \\ &\text{Line } m \text{ passes through } A \text{ and } B \\ &A \text{ and } B \text{ are distinct points} \\ &\text{Therefore, line } l \text{ and line } m \text{ must be the same line} \end{aligned}

This contradicts the given fact that lines ll and mm are distinct.

Hence, our assumption was false.

Answer

Hence, proved by contradiction that two distinct lines in a plane cannot intersect in more than one point.

Common Mistakes
  • Overlooking the Axiom: Forgetting to explicitly cite the geometric axiom that only one line passes through two distinct points.
  • Not Assuming Distinct Points: In a proof by contradiction, assuming they intersect at "more than one point" must be translated to assuming at least two distinct points, say AA and BB.

More questions in A1.6

Q1

Suppose a+b=c+da + b = c + d, and a<ca < c. Use proof by contradiction to show b>db > d.

Q2

Let rr be a rational number and xx be an irrational number. Use proof by contradiction to show that r+xr + x is an irrational number.

Q3

Use proof by contradiction to prove that if for an integer aa, a2a^2 is even, then so is aa.

[Hint : Assume aa is not even, that is, it is of the form 2n+12n + 1, for some integer nn, and then proceed.]

Q4

Use proof by contradiction to prove that if for an integer aa, a2a^2 is divisible by 3, then aa is divisible by 3.

Q5

Use proof by contradiction to show that there is no value of nn for which 6n6^n ends with the digit zero.

Q6

Prove by contradiction that two distinct lines in a plane cannot intersect in more than one point.

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