Purpose: Let’s Define Some Terms!!
CONVERSE
If and only if
Implies
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)
Algebra
If 3( x  2)  42 then x  16 (True)
If x =16
then 3( x  2)  42
(True)
What is the Converse of a Theorem?
Consider the sentence below:
If
Then
the angles of the shape add up to 180o
the shape is a triangle.
To make the converse statement swap around the
parts of the statement in the green boxes above
This is the
converse
statement
(TRUE)
Not all converse statements are true.
Consider the sentence below:
If
aa shape
shape isis aa square.
square
Then
the
theangles
anglesadd
addup
upto
to360
360oo
Now make the converse statement.
Can you think of a shape with angles of 360o which is not a square ?
Any closed quadrilateral.
Is the Converse True or False?
(1) If a triangle has three equal sides then it has True
three equal angles.
(2) If a number is even then the number divides
True
by two exactly.
(3) If a shape is a square then the shape has
False
parallel sides.
(4) If you have thrown a three and a four then False
your total score is seven with a die.
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.
Reductio ad Absurdum: Proof by Contradiction
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.
Proof by Contradiction : Algebra
Theorem: There is no solution to the equation
x 2  3x  1
x
x3
Proof: (By Contradiction)
Suppose there IS a solution. Call it p
p2  3 p  1

 p
p3
 p 2  3 p  1  p ( p  3)
 p2  3 p  1  p2  3 p
1 0
............Impossible
Hence, no solution exists. Q.E.D.
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??
i.e. If x  y  60 then either x  30 or y  30
Assume neither shirt costs more than €30
x ≤ 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
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 900 + 900 + ∠C = 1800
∴ ∠C = 00 which is a contradiction
∴ ∠A and ∠B cannot both be right angles
⇒ A triangle can have at most one right angle
Proof by Contradiction: The Square Root of 2 is Irrational
2
To prove that 2 is irrational
Assume the contrary: 2 is rational
1
i.e. there exists integers p and q with no common factors such that:
p
 2
q
(Square both sides)
p2
 2 2
q
(Multiply both sides by
q 2)
 p 2  2q 2
 p 2 is even
(......it’s a multiple of 2)
 p is even
(......even2 = even)
1
 p is even
p
 2
q
 p  2k for some k
If p  2k
p2
 2 2
q
 p  2q
2
 p 2  2q 2 becomes (2k ) 2  2q 2
2
.
 p is even
2
 p is even
 4k 2  2q 2
(Divide both sides by 2)
 2k 2  q 2
Then similarly q  2m for some m
p 2k
p
 
 has a factor of 2 in common.
q 2m q
This contradicts the original assumption.
2 is irrational
Q.E.D.
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