Chapter 2

Proof

consists of a series of steps, each of which follows logically from assumptions, or from previously proven statements, whose final step should result in the statement of the theorem being proven.

Existential Instantiation

is a law of logic that says if an object is known to exist, then that object can be given a name, as long as the name is not currently being used to denote something else. The definitions of odd and even numbers, rational numbers, and divides all use existential instantiation. If n is an odd integer, then n is equal to two times an integer plus 1. That is, n = 2k+1, for some integer k. Giving the integer k a name is an example of existential instantiation.

Irrational Number

is a real number that is not rational. Note that the definition implies that every real number is either rational or irrational but not both. Therefore if x is a real number that is not irrational, then x is rational.

Composite

is a whole number that can be divided evenly by itself and by 1, as well as by one or more other whole numbers. An integer is a composite iff n > 1, and there is an integer m such that 1 < m < n and m|n.

De Morgan's laws

(A and B) is false if and only if A is false or B is false. (A or B) is false if and only if A is false and B is false.

Logical Proof

- We want to prove that a statement, q, is true - ... therefore q is true - What do we know? - p₁, p₂, p₃, ... , pₙ are all true - What can we conclude from what we know? - apply rules of Inference Establishing that a logical argument is valid(or invalid) is a logical proof

By definition

A fact that is known because of a definition, can be started with the phrase "By definition". For example: "The integer m is even. By definition, m = 2k for some integer k."

By assumption

A fact that is known because of an assumption, can be started with the phrase "By assumption". For example: "By assumption, x is positive. Therefore x >0."

Rational

A number r is rational if there exists integers x and y such that y ≠ 0 and r = x/y

Write proofs in complete sentences.

A proof should read like English text. In mathematical proofs, English sentences often contain mathematical expressions but those should read naturally as part of the sentence. For example: "If x is an integer that is greater than 0, then x ≥ 1."

Existence Proof

A proof that shows that an existential statement is true.

Universal Generalization

A proof that uses universal generalization to prove a universal statement names an arbitrary object in the domain and proves the statement for that object.

Negative

A real number is negative iff x < 0.

Positive

A real number is positive iff x > 0.

Non-Negative

A real number x is non-negative iff x ≥ 0.

Non-Positive

A real number x is non-positive iff x ≤ 0.

Thus and Therefore

A statement that follows from the previous statement or previous few statements, can be started with "Thus" or "Therefore". - n and m are integers. Therefore, - n+m is also an integer. n is a positive integer. Thus, n ≥ 1. Other words that serve the same purpose are "it follows that", "then", "hence".

Prime

An integer n is prime if the only numbers that divide into it evenly are 1 and itself. An integer n is prime iff n > 1, and for every positive integer m, if m divides n, then m = 1 or m = n.

Divides

An integer x divides an integer y iff x ≠ 0 and y = kx for some integer k. The fact that x divides y is denoted by x|y. If x does not divide y, then that fact is denoted x∤y. If x divides y, then y is said to be a multiple of x, and x is a factor or divisor of y.

Even

An integer x is even if there is an integer k such that x = 2k.

Odd

An integer x is odd if there is an integer k such that x = 2k + 1.

When to use a direct proof vs. a proof by contrapositive

Deciding whether to prove a conditional statement using a direct proof or a proof by contrapositive often involves some trial and error. The decision should be based on whether the hypothesis or the negation of the conclusion provides a more useful assumption to work with. Consider the statement:

Mathematical Systems

Definition: What are the concepts - a proposition is a statement that is either true or false. - two angles are supplementary if their sum is 180°. Theorem: - a statement that can be proven to be true. Proof: - A valid argument that established the truth of a theorem. Lemma: - A theorem that is not important on its own but is useful in proving another theorem. Corollary: - A theorem that follows directly from a theorem.

Introduce each variable when the variable is used for the first time.

Here are some examples of the introduction of a new variable: - "Let x be a positive integer." - "Since we know that m divides n, there is an integer k such that n = km." This sentence introduces the variable k. Variables m and n should already have been introduced. - "Let s be the average of x and y: s = (x+y)/2." This sentence introduces the variable s. Variables x and y should already have been introduced.

Four Types of Methods

Direct method: Here we want to show If p, then q(p → q) so the method will be write assume p, then show that q logically follows. Contrapositive: Here we will show that p → q by showing ∼q → ∼p which is logically equivalent. Assume ¬q. Follow a series of steps to conclude ¬p. Contradiction: Here to prove p, we have to prove that ∼p will produce a wrong result and so p must be true. Prove by cases: Here we will establish different types of possible cases to prove p, then by studying the results of all the cases we will show p is true.

Common Mistakes in Proofs

Generalizing from examples Skipping steps Circular reasoning Assuming facts that have not yet been proven

Proof By Contrapositive

Here we will show that p → q by showing ∼q → ∼p which is logically equivalent. Assume ¬q. Follow a series of steps to conclude ¬p.

Since

If a statement depends on a fact that appeared earlier in the proof or in the assumptions of the theorem, it can be helpful to remind the reader of that fact before the statement. The phrase "because we know that" can serve the same purpose. For example, assuming that the facts x > 0 and y > z have been established earlier, a proof could say: - "Since x > 0 and y > z, then xy > xz." - "Because we know that x > 0 and y > z, then xy > xz."

Proof by Exhaustion

If the domain of a universal statement is small, it may be easiest to prove the statement by checking each element individually. A proof of this kind is called a proof of exhaustion.

Direct Proof

In a direct proof of a conditional statement, the hypothesis p is assumed to be true and the conclusion c is proven as a direct result of the assumption. A way of showing the truth or falsehood of a given statement by a straightforward combination of established facts without making any further assumptions. Assume p. Follow a series of steps to conclude q.

A block of equations should be introduced with English text and each step that does not follow from algebra should be justified.

In the example below the facts that n = k+1 and m ≥ 0 should be previously stated assumptions in the proof or previously proven facts: Plugging in n = k + 1 into n² : n² = (k + 1)² = k² + 2k + 1 ≤ k² + 2k + 1 + m, because m ≥ 0. If the justification for a step does not fit easily on the line of the equation, the justification can be provided right after the block of equations.

Indicate when the proof starts and ends.

In this material, every proof begins with the word Proof: and ends with the symbol ■.

Let

New variable names are often introduced with the word "let". For example, "Let x be a positive integer".

Proof by Circular Argument

Prove that n propositions are equivalent. Either they are all true or all false. p₁ → p₂ p₂ → p₃ ... pₙ-₁ → pₙ pₙ → p₁

Mathematical Proofs

Similar to logical proofs - premises - conclusion Different from logical proofs - The rules of Inference are not explicitly stated - Easier to develop, understand and explain

Proofs by Contrapositive of Conditional Statements with Multiple Hypotheses

Some conditional statements have more than one hypothesis. In a proof by contrapositive it is only necessary to show that one of the hypotheses is false, assuming that the rest of the hypotheses are true and the conclusion is false. For example, consider the conditional statement: If H1 and H2 are both true then C is true. The contrapositive of this conditional statement is: If C is false, then it cannot be the case that H1 and H2 are both true. By De Morgan's law, the statement is equivalent to: If C is false, then H1 is false or H2 is false. which is in turn equivalent to: If C is false and H1 is true, then H2 is false. Sometimes using this form of the statement can make it easier to write the proof. A proof by contrapositive in this form would start as follows: Assume that C is false and H1 is true. We shall show that H2 is false. It is also valid to swap the roles of H1 and H2 and start the proof with: Assume that C is false and H2 is true. We shall show that H1 is false.

Gives and Yields

Sometimes a proof is clearer if even an algebraic step is justified. The words "gives" and "yields" are useful to say that one equation or inequality follows from another. - Multiplying both sides of the inequality x > y by 2, gives 2x > 2y. - Substituting m = 2k into m² yields (2k)² - Since z > 0, we can multiply both sides of the inequality x > y by z to get xz > yz.

In other words

Sometimes it is useful to rephrase a statement in a more specific way. The phrase "in other words" is useful in this context. For example: "We must show that the average of x and y is positive. In other words, we must show that (x+y)/2 > 0."

Give the reader a roadmap of what has been shown, what is assumed, and where the proof is going.

The beginning of a proof should always state what facts are assumed. It can also be helpful to inform the reader what will be proven in the proof. If a proof is long, it is helpful to indicate at one or more points in the middle what has been proven and what has yet to be proven. For example: "We have shown that n is a positive integer. Now we must establish that n is composite."

Allowed Assumptions in Proofs

The rules of algebra. For example if x, y, and z are real numbers and x = y, then x+z = y+z. The set of integers is closed under addition, multiplication, and subtraction. In other words, sums, products, and differences of integers are also integers. Every integer is either even or odd. If x is an integer, there is no integer between x and x+1. In particular, there is no integer between 0 and 1. The relative order of any two real numbers. For example 1/2 < 1 or 4.2 ≥ 3.7. The square of any real number is greater than or equal to 0.

Suppose

The word "suppose" can also be used to introduce a new variable. For example: "Suppose that x is a positive integer". Suppose is also used to introduce a new assumption, as in: "Suppose that x is odd", assuming that x has already been introduced as an integer earlier in the proof.

Inequalities

a mathematical sentence involving <, >, =, or ≤, ≥. If x and c are real numbers, then exactly one of the following statements is true: x < c x = c x > c The values of x and c can also be related using the symbols ≤ or ≥. x ≤ c iff x = c or x < c x is at least c or x is greater than or equal to c. x ≥ c iff x = c or x > c x is at most c or x is less than or equal to c.

Perfect Square

a number n is a perfect square if n = k² for some integer k.

Theorem

a statement that can be proven to be true.

Vacuous Proof

an implication happens when the hypothesis of the implication is always false.

Constructive proof of existence

gives a specific example of an element in the domain or a set of directions to construct an element in the domain that had the required properties.

Factor

if a divides b, then a is a factor of b.

Counter Example

is an assignment of values to variables that shows that a universal statement is false. A counterexample for a conditional statement must satisfy all the hypotheses and contradict the conclusion.

Without Loss of Generality

is used in mathematical proofs to narrow the scope of a proof to one special case in situations when the proof can be easily adapted to apply to the general case. (sometimes abbreviated WLOG or w.l.o.g.)

Parity

of a number is whether the number is odd or even. If two numbers are both even or both odd, the two numbers have the same parity. If one number is off and the other is even, then the two numbers have opposite parity.

Proof by Cases

of a universal statement such as ∀x P(x) breaks the domain for the variable x into different classes and gives a different proof for each class. The proof for each class is called a case. In a proof by cases, the cases are numbered, and each case begins with "Case n:", where n is the number of that case. The number is followed by a statement of the assumptions for that case. r₁(x) → q(x) r₂(x) → q(x) ... rₙ(x) → q(x)

Proof by Contrapositive

proves a conditional theorem of the form p → c by showing that the contrapositive ¬c → ¬p is true. In other words, ¬c is assumed to be true and ¬p is proven as a result of ¬c. show ∼hypothesis ... ... ... Assume ∼conclusion

Non-constructive proof of existence

proves that an element with the required properties exist without giving a specific example.

Trivial Proof

refers to a statement involving a material implication P → Q, where the consequent, Q, is always true.

Proof by Contradiction

starts by assuming that the theorem is false and then shows that some logical inconsistency arises as a result of the assumption. The reasoning behind proof by contradiction is that if the assumption that the theorem is false leads to a conclusion which cannot be true, then the theorem must be true. A proof by contradiction is sometimes called an indirect proof. Assume ∼theorem ... ... ... Show inconsistency Assume p ∧ ¬q is true. Follow a series of logical steps to conclude r ∧ ¬r for some proposition r.

Axioms

statements assumed to be true, may be used to prove a theorem.

Consecutive

two integers are consecutive if one of the numbers is equal to 1 plus the other number. y = x + 1

Proof Methods

∃x∈D,p(x) - constructive method - non-constructive method ∀x∈D,p(x) - Method of Exhaustion: can be used only if D is finite - When p(x) is of the form r(x) → q(x): vacuous proof (show that r(x) is always false), trivial proof (show that q(x) is always true, direct proof (choose x, an arbitrary member of D, such that r(x) is true. Show that q(x) is true).