1. Purpose: Let’s Define Some Terms!! Proof by Contradiction
CONVERSE
Corollary
Proof by Contradiction
Terms to be defined in syllabus. In booklet as definitions but understanding is key. Have shown up in exams already.
If and only if
Axiom
Implies

2. Converse of a Statement
Statement: If Rex is a dog
then Rex is a mammal. (True)
Converse: If Rex is a mammal
then Rex is a Dog (False)
Converse of a statement first.
Could be introduced to 1st years when solving equations-preparing groundwork again for later.

3. What is the Converse of a Theorem?
Consider the sentence below: If If
the angles of the shape add up to 180o
the angles of the shape add up to 180o
Then
Then
the shape is a triangle.
the shape is a triangle.
To make the converse statement swap around the
parts of the statement in the green boxes above
If Then approach could be taken when introducing all theorems
This is the converse statement
(TRUE)

4. Not all converse statements are true.
Consider the sentence below: If If
a shape is a square.
a shape is a square
Then
Then
the angles add up to 360o
the angles add up to 360o
Now make the converse statement.
Ask teachers to get converse of some theorems on the course themselves and say whether true or not
Can you think of a shape with angles of 360o which is not a square ?
Any closed quadrilateral.

5. Is the Converse True or False?
(1) If a triangle has three equal sides then it has three equal angles.
True
(2) If a number is even then the number divides by two exactly.
True
(3) If a shape is a square then the shape has parallel sides.
False
(4) If you have thrown a three and a four then your total score is seven with a die.
False

6. Introducing Indirect Proof: Leinster game?
Paul and Mike are driving past the Aviva Stadium. The floodlights are on.
Paul: Are Leinster playing tonight?
Mike: I don’t think so. If a game were being played right now we would see or hear a big crowd but the stands are empty and there isn’t any noise.
Mike is arguing that there is no game going on by supposing that there is a game in the stadium. This leads him to his contradiction and therefore his supposition must be incorrect.
Note on proof by contradiction: Assume a statement is not true and show that this assumption leads to a contradiction – called reduction as absurdum (reduction to absurdity) in Latin)
Reductio ad Absurdum: Proof by Contradiction

7. Introducing Indirect Proof
Sarah left her house at 9:30 AM and arrived at her aunts house 80 miles away at 10:30 AM.
Use an indirect proof to show that Sarah exceeded the 55 mph speed limit.

8. Proof by Contradiction : Algebra
Again students need to meet these ideas way before Geometry.

9. Proof by Contradiction: Inequalities
If Tim buys two shirts for just over €60, can you prove that at least one of the shirts cost more than €30??

10. QED x ≤ 30 y ≤ 30 x + y ≤ 60 Assume neither shirt costs more than €30
y ≤ 30
x + y ≤ 60
This is a contradiction since we know Tim spent more than
€60
Our original assumption must be false
At least one of the shirts had to have cost more than €30
QED

11. Geometry : Proof by Contradiction
Triangle ABC has no more than one right angle.
Can you complete a proof by contradiction for this statement?
Assume ∠A and ∠B are right angles
We know ∠A + ∠B + ∠C = 1800
By substitution ∠C = 1800
∴ ∠C = 00 which is a contradiction
∴ ∠A and ∠B cannot both be right angles
Ask teachers for their own proof by contradiction examples
⇒ A triangle can have at most one right angle

12. Proof by Contradiction: The Square Root of 2 is Irrational2 1
To prove that 2 is irrational
Assume the contrary: 2 is rational
i.e. there exists integers p and q with no common factors such that:
(Square both sides)
(Multiply both sides by )
Considered one of the classic proofs by contradiction.
Proof attributed to Aristotle.
Run through in class then get students to try themselves.
(......it’s a multiple of 2)
(......even2 = even)

13. This contradicts the original assumption.
(Divide both sides by 2) .
This contradicts the original assumption.
2 is irrational Q.E.D.

14. Click for link to external website Proof sorting activity.
Proof could be asked like this in the exam.