1.4 KiB
ECE 108: Discrete Math 1
An axiom is a defined core assumption held to be true.
!!! example True is not false.
A theorem is a true statement derived from axioms via logic or other theorems.
!!! example True or false is true.
A proposition/statement must be able to have the property that it is exclusively true or false.
!!! example The square root of 2 is a rational number.
An open sentence becomes a proposition if a value is assigned to the variable.
!!! example \(x^2-x\geq 0\)
Truth tables
A truth table lists all possible truth values of a proposition, containing independent statement variables.
!!! example | p | q | p and q | | — | — | — | | T | T | T | | T | F | F | | F | T | F | | F | F | F |
Logical operators
!!! definition - A compound statement is composed of component statements joined by logical operators AND and OR.
The negation operator is equivalent to logical NOT.
\[\neg p\]
The conjunction operaetor is equivalent to logical AND.
\[p\wedge q\]
The disjunction operator is equivalent to logical OR.
\[p\vee q\]
The implication sign requires that if \(p\) is true, \(q\) is true, such that \(p\) implies \(q\). The first symbol is the hypothesis and the second symbol is the conclusion.
\[p\implies q\]
| \(p\) | \(q\) | \(p\implies q\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | F |