Math 9th grade honors - truth tables, statments and truth values.

statment

a declarative sentence that is either true or false, but not both

how do you write statements

let p = x (for example)

open statment

a sentence that cannot be described as true or false

Domain

the replacement set

solution set

the set of elements from the domian that make the statment true

negation of a statement

The statement "Not P", written ~ P.
If the statement is" I like eating apples". Then the negation should be "I don't like eating apples"

compound statement

connective words or phrases connect two or more statemnts, two or more statements joined by the word "and" or "or"

Conjunction

a compound statement formed by joining two or more statements with the word "and", each part is called a conjunct, the symbol for it is "^" - so p^q would be read as p and q

p^q

p and q, each conjunct must be true for the conjuncition to be true

True Conjunction

A conjunction is true whenever and wheneverr if both conjunct are true

disjunction

a compound statement formed by joining two or more statements with the word "or", each part is called a disjunct and it is represneted by "v" so "p v q" is the same as p or q

a true disjunction

a disjunction is true when at least one disjunct is true.
pvq - so p can be true and q can be false but the disjunction is true

conditional statement

a statement that can be written in if-then form, p->q the two parts are the antecedent(p) and the consequent(q), in conditional statements they are true when both,q or neither are true

other ways to say conditional statements

if p then q if x+5=8 then x=3
p implies q x+5=8 implies x=3
q if p x=3 if x+5=8
p is sufficient for q x+5=8 is sufficient for x=3
q is necessary for p x=3 is necessary for x+5=8

tautology

when an argumnet (compound statment) is true for every possibility b/c the last column is always true

contradiciton

when an arguement(compound statemnet) is false because every possibilty b/c the last column is always false

if you live in Scarsdale, then you live in westchester (p->q)
find the converse

if you live in westchester then you live in scarsdale
(q->p)

if you live in scarsdale, then you live in westchester(p->q)
find the inverse

if you don't live in scarsdale, then you don't live in Westchester
(~p->~q)

if you live in Scarsdale, then you live in westchester (p->q)
find the contrapositive

if you don't live in westchester then you don't live in scarsdale (~q->~p)

the contrapositive

(~q->~p), eqaul to the conditional statmenet

the inverse

(~p->~q) not equal to condiitonal statement

converse

(q->p) not equal to conditional statment

Biconditional

If (p->q) ^ (q->p) is a true conjunction, then the conjunctions is a biconditional, written as p iff q or
(p

importance of biconditionals

they establish logical equivalents and are often used in mathematical definitions