A Proof is a valid argument that establishes the truth of a statement or proposition.

An Axiom is an accepted truth which can be used to help construct a logical argument

Conjecture is a statement proposed to be true (to be proven or disproven)

Theorem is a statement that can be shown to be true

Corollary A theorem that can be established directly (easily) from a theorem that has been proven.

Types of proofs (and proof strategies) that we will discuss in this section include the following:

Direct Proof-

Proof by Contraposition-

Proof by Contradiction-

Existential Proof-

Proof by cases-

Proof of Logical Equivalences-

Direct Proofs: A proof strategy which verifies the truth of an implication by directly constructing an argument of that form.

Conjecture is in the form of an implication, ie, p -> q or similarly

Observe: a logical argument is by definition in the form of an implication. (In general, we must have some premise in order to reach any conclusion.) Thus we can "directly" show the truth of an implication by constructing an argument of the corresponding form

Direct Proof Strategy: If attempting a direct proof, simply show that q follows from p via logical argument. That is, build a logical argument of the form p -> q.

The "formality" spectrum for proofs

Organization is key!

Good

Bad!

Proof template (basic format for proofs)

Clearly and logically state the conjecture -- the statement to be 1.

Intro to ProofsSunday, August 20, 2017 1:47 PM

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Clearly and logically state the conjecture -- the statement to be proven.

1.

Furthermore: Define all variables used, predicates, domains, etc.

a.

State (and label) all facts, axioms, laws of inference, etc. to be used in proof (so that you can simply refer to them by name or label in the proof)

b.

Declare proof method to be employed. Summarize proof strategy.2.

Proof: construct the appropriate logical argument given your proof strategy. Clearly and methodically justify each step of the proof.

3.

End the proof. Summarize the proof and conclude the truth (or falsehood) of the conjecture. Mark the end of the proof.

4.

Example 1.26: Direct Proof Example

Show that if n is odd, then 3n is odd.

You may use the following facts in the proof:

F.1: an integer n is odd iff there exists an integer m, such that n = 2m+1. (Definition of an odd number)

F.2 if m is an integer, then cm + k is also an integer, for integers c and k

TIPS:

Follow the steps and format for proofs. Always start by clearly identifying the conjecture and writing it out in the appropriate propositional or predicate form.

When it is time to construct the logical argument; always plan out your path before beginning

Step 1:

Step 2:

Step 3: (Proof!)

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Step 4: (Concluding Remarks)

Another Example of Direct Proof (from Hammack)

Illustrative example of planning the proof journey or path!

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Some proofs are better justified "indirectly", ie, building a logical argument in a form other than the conjecture, but the truth of the argument implies the truth of the conjecture.

For example: I can show that statement p -> q is true, by showing that ~q -> ~p (the contrapositive) is true.

Proof by Contraposition: To prove the truth of some implication, build a logical argument of the form of the contrapositive, thus showing the truth of the contrapositive which is logically equivalent to the original implication.

Example 1.27. Show that if 5n is odd, then n is odd.Proof Strategy: Try direct proof first. It is often the easiest to construct. If that fails try an indirect proof.

Lets try to use a direct proof first …

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Proof by Contradiction. (an indirect proof strategy based on Modus Tollens.) Idea: Suppose we wish to prove that some logical

statement s is true. Now suppose we can find some p such that ~s -> p. Further assume that p is a contradiction (always false). Since ~s -> p and ~p are assumed true, then s must be true.

Steps in proof by contradiction. Given our discussion above, we can construct an argument as follows: To prove the truth of some logical statement s, construct a logical argument of the form ~s -> F. In other words, show that negating your conjecture leads to a contradiction.

Example 1.28: Prove - There is no greatest integer.

Use Fact F.2: if n is an integer then cn + k is an integer for all integers c and k.

First, lets translate to logic

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Proof of Logical Equivalence.For conjectures of the form: p <-> q

Truth Table (of course)1.Sequential substitution of logical equivalences 2.

Note p <-> q is logically equivalent to p ->q AND q -> p.

a.

Simply show that both implications are true. b.

Proof by cases strategy:3.

Some Approaches

Proof by Example.For quantified conjectures of the form:

Existential Proof: Prove the truth of an existentially quantified predicate via an example. -

Show that a universally quantified statement is false○

Counter Example: Prove the falsehood a universally quantified predicate with an example to the contrary.-

Thus finding a single example when the conjecture is true is sufficient.

Example 1.29. Counter ExampleTip: Before beginning a proof, it is a good idea to try a few examples to identify if the statement is true or false for those examples. This approach is clear if your conjecture is an existentially quantified predicate.

HOWEVER, it may be a good idea to perform this preliminary investigation for conjectures that are universally quantified. WHY?

You may get lucky and find a counter example! Or the result of this preliminary investigation may give you some insight into the overall proof strategy.

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Proof by cases and exhaustive proofs

For conjectures in the form of an implication:And it is convenient (for the purposes of completing the proof) to "split" our premise into cases:

With quantified statements the cases may be simply partitioning the domain into multiple sub-domains.

Example 1.31

1.30. Existential Proof

As you may have noticed, there are many

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As you may have noticed, there are many proof strategies! How can you determine which strategy to use to prove or disprove a conjecture?

Formalize and identify the form the conjecture. Given the form, you can likely eliminate many potential strategies

1.

Try a few examples. If you have an existential form, this strategy is clear. If your form is universally quantified, you may get lucky and find counter example.

2.

Try a direct proof.3.Try an indirect proof. 4.

Proof Strategy

Proof by Resolution: The repeated use of Resolution within the logical argument constructed in a proof by contradiction (proof by contraposition).

Convert your premises into conjunctive normal form (product of sums). Thus each premise should be a disjunction of a simple proposition or the negation of a simple proposition.

1.

Proceed with a proof by contradiction, where each step of constructing the appropriately formed logical argument consists of applying resolution to two previously justified statements. That is, negate the conjecture, add it to the set of premises; then attempt to derive a contradiction.

2.

Steps

See HW

Example 1.32. Proof by Resolution

Show whether the following argument is valid

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Show whether the following argument is valid

First: Convert the premises (knowledge base) into CNF form

Next, Show that negating the consequence (query) results in a contradiction.

Mistakes in Proofs. Be clear. Stay Organized!

Try this at home!

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Open Problems. The truth of some conjectures are unknown. There exist many open problems in mathematics and computer science. We will discuss a few as we progress through the course.

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