logic and deductive reasoning

factoring: a³+b³

factoring: a³+b³

(a+b)(a²-ab+b²)

factoring: a³-b³

(a-b)(a²+ab+b²)

conjunction

a compound statement formed by joining two statements with the word "and." each statement is called a conjunct.

disjunction

a compound statement formed by joining two sentences with the word "or." each statement is called a disjunct.

conditional statements (6 ways to write them)

if p (antecedent), then q (consequent). can also be written as: q if p, p implies q, p only if q, p is sufficient for q, q is necessary for p.

tautology

when a statement is always true

bi-conditional statement

when (p→q)Ʌ(q→p) is true. can be stated as p if and only if q (p iff q). bi-conditionals are important for establishing logical equivalence and are often used in mathematical definitions.

law of the contrapositive

(p→q)↔(~q→~p)

law of double negation

p↔~(~p) (p is logically equivalent to not not p)

definition of a true conjunction

(pɅq)→q; if a conjunction is true, then each conjunct is true

law of disjunctive inference

[(pVq)Ʌ~q]→p; if a disjunction is true and one of it's disjuncts is false, than the other disjunct must be true

multiplying two monomials

to multiply a monomial by a monomial, we multiply the numerical factors/co-efficients and then multiply the variable factors

multiplying a polynomial and a monomial

to multiply a polynomial by a monomial, we multiply each term of the polynomial by the monomial

multiplying a polynomial by a polynomial

to multiply a polynomial by a polynomial, we multiply each term of one polynomial by each term of the other polynomial

the square of a binomial

the square of a binomial is the square of the first term, twice the product of its two terms, //plus// the square of the last term

multiplying the sum and difference of the same two terms

the product of the sum and difference of the same two terms is the square of the first term minus the square of the second term

conjunctive addition

if given p and given q, then the conclusion pɅq can be drawn

disjunctive addition

p→(pVq); given p implies pVq is true

law of detachment

[(p→q)Ʌp]→q; if the antecedent of a true conditional is true, then the consequent must be true

law of modus tollens

[(p→q)Ʌ~q]→~p; if the consequent of a true conditional is false then the antecedent is always false

law of syllogism

[(p→q)Ʌ(q→r)]→(p→r); if p→q is true and q→r is true, then p→r is true

demorgans law (negation of a conjunction)

~(pɅq)↔(~pV~q); the negation of a conjunction is logically equivalent to a disjunction whose statements are the negation of it's conjuncts

demorgans law (negation of a disjunction)

~(pVq)↔(~pɅ~q); the negation of a disjunction is logically equivalent to a conjunction whose statements are the negations of its disjuncts

disjunctive equivalent of a conditional

(p→q)↔(~pVq); every conditional statement has a logically equivalent disjunction whose disjuncts are the negation of the antecedent, and the consequent

the negation of a conditional

~(p→q)↔(pɅ~q); the negation of a conditional is logically equivalent to a conjunction whose conjuncts are the antecedent followed by the negation of the consequent

the negation of a bi-conditional

in p↔q, p and q must have the same truth value, therefore the negation of ~(p↔q) is ~p↔q or p↔~q