Download § 2.1 Mathematical Systems, Direct Proofs and Counterexamples

§ 2.1 Mathematical Systems, Direct Proofs
and Counterexamples
Types of Mathematical Statements
1
Axiom - statement that is accepted to be true without proof
Types of Mathematical Statements
1
Axiom - statement that is accepted to be true without proof
Example: Euclid’s axioms, the first of which is “A straight line
segment can be drawn joining any two points.”
Types of Mathematical Statements
1
Axiom - statement that is accepted to be true without proof
Example: Euclid’s axioms, the first of which is “A straight line
segment can be drawn joining any two points.”
2
Theorem - a mathematical statement whose truth can be verified,
although mathematician usually reserve this term for important
or interesting.
Types of Mathematical Statements
1
Axiom - statement that is accepted to be true without proof
Example: Euclid’s axioms, the first of which is “A straight line
segment can be drawn joining any two points.”
2
Theorem - a mathematical statement whose truth can be verified,
although mathematician usually reserve this term for important
or interesting.
3
Corollary - a mathematical statement that is a consequence of a
theorem or an axiom.
Types of Mathematical Statements
1
Axiom - statement that is accepted to be true without proof
Example: Euclid’s axioms, the first of which is “A straight line
segment can be drawn joining any two points.”
2
Theorem - a mathematical statement whose truth can be verified,
although mathematician usually reserve this term for important
or interesting.
3
Corollary - a mathematical statement that is a consequence of a
theorem or an axiom.
4
Lemma - a mathematical result that is useful in verifying the
truth in another result.
Direct Proofs
Definition
A direct proof is a way of showing the truth or falsehood of a given
statement by a straightforward combination of established facts,
usually existing lemmas and theorems, without making any further
assumptions.
Direct Proofs
Definition
A direct proof is a way of showing the truth or falsehood of a given
statement by a straightforward combination of established facts,
usually existing lemmas and theorems, without making any further
assumptions.
Definition
An indirect proof may begin with certain hypothetical scenarios and
then proceed to eliminate the uncertainties in each of these scenarios
until an inescapable conclusion is forced.
How To Write Proofs: The Basics
1
Always state the assumptions that can be made.
How To Write Proofs: The Basics
1
Always state the assumptions that can be made.
2
Always state what you are trying to prove.
How To Write Proofs: The Basics
1
Always state the assumptions that can be made.
2
Always state what you are trying to prove.
3
Always separate strings of equations - whether a simple
statement or a long series of algebraic steps, separation makes it
easier for the reader to be able to verify the work.
How To Write Proofs: The Basics
1
Always state the assumptions that can be made.
2
Always state what you are trying to prove.
3
Always separate strings of equations - whether a simple
statement or a long series of algebraic steps, separation makes it
easier for the reader to be able to verify the work.
Always state that the proof is complete in some way
4
Use a black box or an open box
How To Write Proofs: The Basics
1
Always state the assumptions that can be made.
2
Always state what you are trying to prove.
3
Always separate strings of equations - whether a simple
statement or a long series of algebraic steps, separation makes it
easier for the reader to be able to verify the work.
Always state that the proof is complete in some way
4
Use a black box or an open box
Write Q.E.D., which is the abbreviation for the Latin phrase
‘quod erat demonstrandum,’, which means ‘which had to be
demonstrated’
How To Write Proofs: The Basics
1
Always state the assumptions that can be made.
2
Always state what you are trying to prove.
3
Always separate strings of equations - whether a simple
statement or a long series of algebraic steps, separation makes it
easier for the reader to be able to verify the work.
Always state that the proof is complete in some way
4
Use a black box or an open box
Write Q.E.D., which is the abbreviation for the Latin phrase
‘quod erat demonstrandum,’, which means ‘which had to be
demonstrated’
5
Never refer to yourself in a proof - if you need to use a pronoun,
use ‘we’
A First Proof
Example
If n is an odd integer then 5n + 3 is an even integer.
A First Proof
Example
If n is an odd integer then 5n + 3 is an even integer.
We need to start with something of the form 2k + 1 and we need to
end with something of the form 2k. So we want to use the definitions
we established above and keep these in mind.
A First Proof
Example
If n is an odd integer then 5n + 3 is an even integer.
We need to start with something of the form 2k + 1 and we need to
end with something of the form 2k. So we want to use the definitions
we established above and keep these in mind.
Proof: Assume that n is an odd integer. We want to show that 5n + 3
is an even integer.
A First Proof
Example
If n is an odd integer then 5n + 3 is an even integer.
We need to start with something of the form 2k + 1 and we need to
end with something of the form 2k. So we want to use the definitions
we established above and keep these in mind.
Proof: Assume that n is an odd integer. We want to show that 5n + 3
is an even integer.
Since n is odd, then ∃ k ∈ Z 3 n = 2k + 1. Now,
5n + 3 = 5(2k + 1) + 3 = (10k + 5) + 3 = 10k + 8 = 2(5k + 4)
Since 5k + 4 ∈ Z, 5n + 3 is an even integer. Q.E.D.
Another Example
Example
If n is an odd integer then n2 is odd.
Another Example
Example
If n is an odd integer then n2 is odd.
Suppose n is an odd integer. Then we can write n in the form
n = 2k + 1 where k ∈ Z.
Another Example
Example
If n is an odd integer then n2 is odd.
Suppose n is an odd integer. Then we can write n in the form
n = 2k + 1 where k ∈ Z. Consider
n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1
Since k is an integer, so is 2k2 + 2k and so we can write n2 in the form
necessary for inclusion in the set of odd integers.Therefore, if n is odd
then n2 is as well. Q.E.D.
Perfect Squares
Example
Every odd integer is the difference of two perfect squares.
Perfect Squares
Example
Every odd integer is the difference of two perfect squares.
Let n be an odd integer. We want to show that n is the difference of
two perfect squares.
Perfect Squares
Example
Every odd integer is the difference of two perfect squares.
Let n be an odd integer. We want to show that n is the difference of
two perfect squares. Since n is odd, ∃ k ∈ Z where n = 2k + 1.
Perfect Squares
Example
Every odd integer is the difference of two perfect squares.
Let n be an odd integer. We want to show that n is the difference of
two perfect squares. Since n is odd, ∃ k ∈ Z where n = 2k + 1. Since
k is an integer, k2 and (k + 1)2 are both perfect squares.
Perfect Squares
Example
Every odd integer is the difference of two perfect squares.
Let n be an odd integer. We want to show that n is the difference of
two perfect squares. Since n is odd, ∃ k ∈ Z where n = 2k + 1. Since
k is an integer, k2 and (k + 1)2 are both perfect squares.
Consider
(k + 1)2 = k2 + 2k + 1
and
k2 + 2k + 1 − k2 = 2k + 1
Therefore, if n is odd then we can express n as the difference of
perfect squares. Q.E.D.
More Odd and Even Integers
Example
If n is an odd integer then 4n3 + 2n − 1 is odd.
More Odd and Even Integers
Example
If n is an odd integer then 4n3 + 2n − 1 is odd.
Proof: Suppose n is an odd integer. We will prove that 4n3 + 2n − 1 is
an odd integer.
More Odd and Even Integers
Example
If n is an odd integer then 4n3 + 2n − 1 is odd.
Proof: Suppose n is an odd integer. We will prove that 4n3 + 2n − 1 is
an odd integer.
If n is odd, then ∃ k ∈ Z 3 n = 2k + 1.
More Odd and Even Integers
Example
If n is an odd integer then 4n3 + 2n − 1 is odd.
Proof: Suppose n is an odd integer. We will prove that 4n3 + 2n − 1 is
an odd integer.
If n is odd, then ∃ k ∈ Z 3 n = 2k + 1. Consider
4n3 + 2n − 1 = 4(2k + 1)3 + 2(2k + 1) − 1
= 4(8k3 + 12k2 + 6k + 1) + 4k + 2 − 1
= 32k3 + 48k2 + 28k + 5
= 2(16k3 + 24k2 + 14k + 2) + 1
More Odd and Even Integers
Example
If n is an odd integer then 4n3 + 2n − 1 is odd.
Proof: Suppose n is an odd integer. We will prove that 4n3 + 2n − 1 is
an odd integer.
If n is odd, then ∃ k ∈ Z 3 n = 2k + 1. Consider
4n3 + 2n − 1 = 4(2k + 1)3 + 2(2k + 1) − 1
= 4(8k3 + 12k2 + 6k + 1) + 4k + 2 − 1
= 32k3 + 48k2 + 28k + 5
= 2(16k3 + 24k2 + 14k + 2) + 1
Since 16k3 + 24k2 + 14k + 2 is an integer, 4n3 + 2n − 1 is odd and
this completes the proof.
A Better Way
Although this is correct, it is not the most efficient proof we could
have written. We could have simply noted that
4n3 + 2n − 1 = 4n3 + 2n − 2 + 1 = 2(2n3 + n − 1) + 1
is odd for all n, making this a trivial proof. It would be better to
rewrite the hypothesis as ‘if n is an integer’.
Intersections and Unions
Example
For all sets X, Y, Z, prove that if X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z
then Y = Z.
Let X, Y, Z be sets. Assume X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z. We
will show that Y and Z are subsets of each other.
Intersections and Unions
Example
For all sets X, Y, Z, prove that if X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z
then Y = Z.
Let X, Y, Z be sets. Assume X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z. We
will show that Y and Z are subsets of each other.
Let y ∈ Y. Then y ∈ X ∪ Y. So, y ∈ X ∪ Z. That means y ∈ X or
y ∈ Z. If y ∈ Z, we are done. So suppose y ∈ X. Then y ∈ X ∩ Y. So,
y ∈ X ∩ Z. Thus both cases lead to y ∈ Z. Hence, Y ⊆ Z.
Intersections and Unions
Example
For all sets X, Y, Z, prove that if X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z
then Y = Z.
Let X, Y, Z be sets. Assume X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z. We
will show that Y and Z are subsets of each other.
Let y ∈ Y. Then y ∈ X ∪ Y. So, y ∈ X ∪ Z. That means y ∈ X or
y ∈ Z. If y ∈ Z, we are done. So suppose y ∈ X. Then y ∈ X ∩ Y. So,
y ∈ X ∩ Z. Thus both cases lead to y ∈ Z. Hence, Y ⊆ Z.
In a symmetric argument, it can be shown by assuming that we have
some z ∈ Z that Z ⊆ Y.
Intersections and Unions
Example
For all sets X, Y, Z, prove that if X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z
then Y = Z.
Let X, Y, Z be sets. Assume X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z. We
will show that Y and Z are subsets of each other.
Let y ∈ Y. Then y ∈ X ∪ Y. So, y ∈ X ∪ Z. That means y ∈ X or
y ∈ Z. If y ∈ Z, we are done. So suppose y ∈ X. Then y ∈ X ∩ Y. So,
y ∈ X ∩ Z. Thus both cases lead to y ∈ Z. Hence, Y ⊆ Z.
In a symmetric argument, it can be shown by assuming that we have
some z ∈ Z that Z ⊆ Y.
Since Y ⊆ Z and Z ⊆ Y, it must be the case that Y = Z. Thus, we
have arrived at the desired result. Q.E.D.
Power Sets
Example
Prove that if A and B are sets then P(A) ∪ P(B) ⊆ P(A ∪ B).
Power Sets
Example
Prove that if A and B are sets then P(A) ∪ P(B) ⊆ P(A ∪ B).
Suppose X ∈ P(A) ∪ P(B). By definition of union, X ∈ P(A) or
X ∈ P(B). Therefore, X ⊆ A or X ⊆ B.
Power Sets
Example
Prove that if A and B are sets then P(A) ∪ P(B) ⊆ P(A ∪ B).
Suppose X ∈ P(A) ∪ P(B). By definition of union, X ∈ P(A) or
X ∈ P(B). Therefore, X ⊆ A or X ⊆ B.
Now, if X ⊆ A, then X ⊆ A ∪ B. So X ∈ P(A ∪ B). But, if X ⊆ B,
then we still have X ⊆ A ∪ B and so again, X ∈ P(A ∪ B). Therefore,
we have shown that if X ∈ P(A) ∪ P(B), then X ∈ P(A ∪ B). Q.E.D.
Rational Numbers
Example
The sum of any two rational number is rational.
Rational Numbers
Example
The sum of any two rational number is rational.
Let r and s be rational numbers. We want to show that r + s is a
rational number.
Rational Numbers
Example
The sum of any two rational number is rational.
Let r and s be rational numbers. We want to show that r + s is a
rational number.
By definition of rational numbers, there are integers a and b with
b 6= 0 such that
a
r=
b
Similarly, there are integers c and d with d 6= 0 such that
s=
c
d
Proof (cont.)
Consider
a c
+
b d
da bc
=
+
db bd
ad
bc
=
+
bd bd
r+s=
Proof (cont.)
Consider
a c
+
b d
da bc
=
+
db bd
ad
bc
=
+
bd bd
r+s=
Let p = ad + bc and let q = bd. By closure of the integers under
addition and multiplication, p and q are integers. Hence,
r+s=
p
q
Therefore, we can write r + s in the appropriate form for inclusion in
the set of rational number. This completes the proof.
Counterexamples
The purpose of counterexamples is to prove a statement is false.
Usually, we use counterexamples to disprove a universally quantified
statement.
Counterexamples
The purpose of counterexamples is to prove a statement is false.
Usually, we use counterexamples to disprove a universally quantified
statement.
For the following, how would we disprove using counterexamples?
1
All professors like pizza.
2
∀ x ((x + 1)2 = x2 + 1)
3
If n is an integer and n2 is divisible by 4, then n is divisible by 4.