a compound statement formed by joining two statements with the word "and." each statement is called a conjunct.
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disjunction
a compound statement formed by joining two sentences with the word "or." each statement is called a disjunct.
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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.
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tautology
when a statement is always true
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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.
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law of the contrapositive
(p→q)↔(~q→~p)
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law of double negation
p↔~(~p) (p is logically equivalent to not not p)
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definition of a true conjunction
(pɅq)→q; if a conjunction is true, then each conjunct is true
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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
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multiplying two monomials
to multiply a monomial by a monomial, we multiply the numerical factors/co-efficients and then multiply the variable factors
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multiplying a polynomial and a monomial
to multiply a polynomial by a monomial, we multiply each term of the polynomial by the monomial
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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
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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
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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
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conjunctive addition
if given p and given q, then the conclusion pɅq can be drawn
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disjunctive addition
p→(pVq); given p implies pVq is true
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law of detachment
[(p→q)Ʌp]→q; if the antecedent of a true conditional is true, then the consequent must be true
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law of modus tollens
[(p→q)Ʌ~q]→~p; if the consequent of a true conditional is false then the antecedent is always false
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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
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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
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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
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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
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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
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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
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