Proof Writing in Discrete Mathematics Test 1

Conditional Statement

If P then Q, denoted as P→Q Only false if P is T and Q is F

Direct Proof (often for proving universal)

1. Express the statement to be proven in a formalized format 2. Write the word Proof 3. Start proof by supposing x∈D and P(x) (hypothesis) - write in grammatically correct sentences, give a reason for each step in the proof 4. Show that the conclusion Q(x) is true by using definitions, previously established results, and the rule for logical inference

Definition of Rational

A number that can be expressed as a quotient of two integers with a nonzero denominator.

Irrational

A real number that is not rational.

Statement (Propositional) Form

Expression made up of statement variables and logical connectives

Proving Existential

Find one example which makes the statement true.

Definition of prime

For all positive integers r and s, if n=rs then either r=1 and s=n or r=n and s=1

Definition of composite

For some positive integers r and s such that n=rs and 1<r<n and 1<s<n

Existential Statement

Given one property that may or may not be true, there is at least one thing for which the property is true ∃y∈D, Q(y) only true if Q(y) is true for at least one y in the domain (D)

Disproving Existential

Must prove that it's negation (which will be an universal) is true

∼P

Not P, the negation of P

P∧Q

P and Q, the conjunction of P and Q only true when P is T and Q is T

Biconditional

P if and only if Q "iff"= if and only if P↔Q only true if P and Q are F or P and Q are T

Hypothesis

P in a conditional statement

P∨Q

P or Q, the disjunction of P and Q true except when P is F and Q is F

Conclusion

Q in a conditional statement

R

Set of all Real Numbers

Set

Set of all elements x in S x∈S

Z

Set of all integers

Q

Set of all rational numbers

De Morgan's Law 1

The negation of P∧Q is ∼P∨∼Q

De Morgan's Law 2

The negation of P∨Q is ∼P∧∼Q

Method of Generalization

To show that every element of a set satisfies a certain property, suppose x is a particular but arbitrarily chosen element of the set and show that x satisfies the property

Truth Values

When a proposition is True or False

Universal Statement

a certain property is true for all elements in a set ∀x∈D, Q(x) only true if Q(x) is true for all x in the domain (D)

Vacuously True

a conditional statement that is true by virtue of the fact that its hypothesis is false

Universal Existential Statement

a statement is universal because the first part says that a certain property is true for all objects of a given type, existential because it's second part exerts the existence of something ex. Every real number has an additive inverse.

Counterexample

a value for x in which Q(x) is false

Definition of even

an integer is even if it is equal to twice some integer even= 2k

Definition of odd

an integer that is one more than twice some integer odd= 2k+1

Truth Table

displays the truth values that correspond to all possible combinations of truth values for its component statement variables

Disproving Universal

find a counterexample and show how it does not satisfy the claim

Zero Product Property

if neither of two real numbers is not zero, then their product is also not zero.

Necessary Condition

if not r then not s if r does not occur, s cannot occur either Q

Sufficient Condition

if r then s r is enough to guarantee the occurrence of s P

Logically Equivalent

if two statements have the same truth values

Contrapositive of P→Q

negate and switch both terms ∼Q→∼P logically equivalent to P→Q

Inverse of P→Q

negate both sides ∼P→∼Q not logically equivalent to P→Q

Statement (Proposition)

sentence that is true or false but not both

Contradiction

statement form that is always false

Tautology

statement form that is always true

Existential Universal Statement

statement that is existential because its first part asserts that a certain object exists and its universal because the second part says that the object satisfies a certain property for all things of a certain kind Ex. there is a positive integer that is less than or equal to every positive integer

Converse of P→Q

switch terms P→Q not logically equivalent to P→Q

Negation of a Conditional

∼(P→Q) = P ∧∼Q

Negation of a Universal

∼(∀x∈D, Q(x)) = ∃x∈D, ∼Q(x)

Negation of an Existential

∼(∃x∈D, Q(x)) = ∀x∈D, ∼Q(x)

Order of Operations

∼, then ∧ and ∨, then →