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Discrete Math - 3.1
Methods of Proof
P. Danziger
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Proof Methods
Note that these methods are only general guidelines, every proof has its own form. The guts of the
proof still needs to be filled in, these guidelines merely provide a possible staring point.
Three Useful rules:
1. Always state what you are trying to prove in symbolic form first.
2. Always go back to the definition of objects under consideration.
3. Never use a variable which has not first been introduced.
Introduce variables with statements like: “Let x ∈ S . . .”
This identifies x as an element of the set S. x is assumed to be a particular, but arbitrary element
of S.
We can assume that x has any of the properties associated with S, but no other properties.
Example 1
Let y ∈ {x ∈ Z | x is even }.
We can now assume that y is even, but nothing else.
With this restriction we can now use y as a particular element of the set.
Thus we may make statements like “2y is even”.
This is understood to hold for every y in the domain of y (even numbers).
However, we may not assume that y is divisible by 4, since this is not a property shared by every
element of the set.
To prove existential statements of the form
“∃ x ∈ S such that P (x)”.
Find a specific instance a ∈ S such that P (a) is true.
This is called proof by construction. In practice we rarely use this method.
To prove universal statements of the form
∀ x ∈ S, P (x).
Start with “Let x ∈ S” (x is an arbitrary but specific element of S, as above).
Show that P (x) follows.
Statements of the form ∼(∃ x ∈ S such that P (x)) can be dealt with by realizing that
∼(∃ x ∈ S such that P (x)) ≡ ∀ x ∈ S, ∼P (x).
Similarly ∼(∀ x ∈ S, P (x)) ≡ ∃ x ∈ S such that ∼P (x).
Thus ∼(∀ x ∈ S, P (x)) can be proven by finding a specific a ∈ S such that P (a) is false – this is
known as a counterexample.
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Discrete Math - 3.1
Methods of Proof
P. Danziger
To prove conditional statements of the form p ⇒ q.
Assume that the predicate p is true and show that, under this assumption, q follows.
Note that p and q may involve quantified variables.
To prove biconditional statements of the form ‘p ⇔ q’.
First prove p ⇒ q.
Then prove p ⇐ q.
Never try to prove a biconditional in one go.
We can mix and match these methods as needed. For example, a proof a statement of the form
below might go as follows.
To Prove ∀ x ∈ S, ∃ y ∈ T such that P (x, y) ⇒ Q(x, y).
Let x ∈ S [x is an arbitrary but specific element of S]
Consider a ∈ T [a is a specific element of S, which you give]
Assume that P (x, a) is true (Assume the predicate)
..
This is the guts of the proof, we may assume that x has
.
any of the properties of S, and that P (x, a) is true
Therefore Q(x, a).
A proof by contradiction works by using the valid form ‘∼ p ⇒ c, Therefore p’.
If we are trying to show the statement p, we assume its negation, ∼ p.
We then show that this assumption implies some statement which we know to be false. We may
then conclude that p is true.
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Assumptions
The basic assumptions of mathematics are called Axioms.
These are statements which are assumed to be true, with no proof. The basic axioms of the real
numbers, as well as some of the basic theorems are given in Appendix A, p. 695 of the book. You
may assume any of the axioms or theorems therein.
Essentially, this means that you may assume the basic rules of algebraic manipulation (Algebra),
where appropriate. But be careful about where these rules are appropriate.
For a, b, c in N, Z, Q or R : a(b + c) = ab + ac. (Algebra)
For a, b, c in N, Z, Q or R : a(b − c) = ab − ac. (Algebra) (Distributive law, Axiom F3 and Theorem
T5)
For a, b, c in N, Z, Q and R : a + b = c ⇔ a = b − c. (Algebra)
For a, b, c in Q and R : ab = c ⇒ a = c/b. (Algebra)
Beware of division in N and Z. Generally we use the notion of divides ( | ) here.
If a and c are in N or Z : ab = c ⇒ a | c (a divides c).
You may assume closure under basic operations as appropriate.
Closure means that the result of an operation is guaranteed to remain in the set. Thus adding two
natural numbers is guaranteed to yield a natural number, so N is closed under ‘+’. On the other
hand subtracting one natural number from another may not give a natural number, so N is not
closed under ‘−’.
• N, Z, Q and R are all closed under addition.
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Discrete Math - 3.1
Methods of Proof
P. Danziger
• N, Z, Q and R are all closed under multiplication.
• Z, Q and R are closed under subtraction.
N is not closed under subtraction.
• Q and R are closed under division.
N and Z are not closed under division.
There are two basic theorems which will be stated without proof. You may assume these theorems,
they are:
• The Unique Factorization Theorem (3.3.3)
• The Quotient Remainder Theorem (3.4.1)
Whenever you use any of these assumptions make sure you attribute it.
(Algebra), (Closure of Z under ×), (Quotient Remainder Theorem), etc.
When doing proofs, keep in mind the goal toward which you are working. Try to develop an
understanding of the problem. Study particular examples and come up with reasons why things
are, or are not true.
Always go back to the definitions of the objects under consideration. This often provides a starting
point, or a clarification of the statement. Note that definitions are always in logical, symbolic form.
Above all proofs should be clear and precise
Do not ramble on, hoping that no one will notice that you don’t know what you are talking about.
Such arguments are called hand waving, due to the large amount of gesticulation that usually
accompanies them.
1.2
Common Mistakes
Read section starting p. 120 of Epp.
1. Proof by Example
A universal statement cannot be proved by an example.
Though you may wish to think about an example to gain insight into a problem an example
never constitutes a proof of a universal statement.
2. Incorrect Variable Use
Don’t use the same variable name for two different things.
m and n are even, so m = 2k, and n = 2k for some integer k.
m and n are not necessarily equal, so this may cause confusion later on.
3. Jumping to a Conclusion
Beware of making unwarranted assumptions (even if they happen to be true).
4. Begging the Question
Don’t assume the result, whether implicitly or explicitly.
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Discrete Math - 3.1
Methods of Proof
P. Danziger
5. Misuse of if
Don’t use if when you mean since or because.
6. Arguing from false premises
Don’t assume something which is not true, then proving the result is always possible. (p ⇒ q
is always true when p is false.)
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Basic Definitions of Number Theory
Definition 2 [Even, Odd, Prime, Composite, Divisible]
• An integer is n is even if and only if n = 2k for some integer k.
Formally:
∀n ∈ Z, n is even ⇔ ∃ k ∈ Z such that n = 2k.
• An integer is n is odd if and only if n = 2k + 1 for some integer k.
Formally:
∀n ∈ Z, n is odd ⇔ ∃ k ∈ Z such that n = 2k + 1.
• An integer n is prime if and only if, n > 1 and for all positive integers r and s, if n = r · s
then either r = 1 or s = 1.
Formally:
∀n ∈ Z, n is prime ⇔ (n > 1 ∧ (∀ r, s ∈ Z+ , n = r · s ⇒ (r = 1 ∨ s = 1))).
• An integer n is composite if and only if n = r · s for some positive integers r and s, with r 6= 1
and s 6= 1.
Formally:
∀n ∈ Z, n is composite ⇔ ∃ r, s ∈ Z+ such that n = r · s ∧ r 6= 1 ∧ s 6= 1.
• Given integers n and d, with d 6= 0 we say that n is divisible by d if there is some integer k
such that n = d · k.
We write d | n to denote ‘d divides n’. Here n and d are integers, and d 6= 0.
Formally:
∀n, d ∈ Z, d | n ⇔ (∃ k ∈ Z such that n = d · k ∧ d 6= 0).
Alternatively, if n is divisible by d, we say that:
• n is a multiple of d;
• d is a factor of n;
• d is a divisor of n;
• d divides n.
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