# First Order Logic Resources

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## Tutorials

1. Rules and terminology by Wolfram Mathworld [FOL]
2. Some [good note]s on how to start with FOL (very programming oriented)
3. Peter Suber's [translating logic into English AND his [glossary] of set theory is immensely helpful for the uninitiated.
1. [PD Magnus] forall x: an introduction to formal logic.
1. Thorough intro, but very "textbookish" below

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## Other Tutorials

Symbol
Name Explanation Examples Unicode
Value
HTML
Entity
LaTeX
symbol
Category

material implication AB means if A is true then B is also true; if A is false then nothing is said about B.

→ may mean the same as ⇒ (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols).

⊃ may mean the same as ⇒ (the symbol may also mean superset).
x = 2  ⇒  x2 = 4 is true, but x2 = 4   ⇒  x = 2 is in general false (since x could be −2). U+21D2

U+2192

U+2283
&rArr;
&rarr;
&sup;
$\displaystyle{ \Rightarrow }$\Rightarrow
$\displaystyle{ \to }$\to
$\displaystyle{ \supset }$\supset
implies; if .. then
propositional logic, Heyting algebra

material equivalence A ⇔ B means A is true if B is true and A is false if B is false. x + 5 = y +2  ⇔  x + 3 = y U+21D4

U+2261

U+2194
&hArr;
&equiv;
&harr;
$\displaystyle{ \Leftrightarrow }$\Leftrightarrow
$\displaystyle{ \equiv }$\equiv
$\displaystyle{ \leftrightarrow }$\leftrightarrow
if and only if; iff
propositional logic
¬

˜

!
negation The statement ¬A is true if and only if A is false.

A slash placed through another operator is the same as "¬" placed in front.
¬(¬A) ⇔ A
x ≠ y  ⇔  ¬(x =  y)
U+00AC

U+02DC
&not;
&tilde;
~
$\displaystyle{ \lnot }$\lnot
$\displaystyle{ \sim }$\sim
not
propositional logic

&
logical conjunction The statement AB is true if A and B are both true; else it is false. n < 4  ∧  n >2  ⇔  n = 3 when n is a natural number. U+2227

U+0026
&and;
&amp;
$\displaystyle{ \land }$\land
\&[1]
and
propositional logic

+
logical disjunction The statement AB is true if A or B (or both) are true; if both are false, the statement is false. n ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number. U+2228 &or; $\displaystyle{ \lor }$\lor
or
propositional logic

exclusive disjunction The statement AB is true when either A or B, but not both, are true. A Template:Unicode B means the same. A) ⊕ A is always true, AA is always false. U+2295

U+22BB
&oplus; $\displaystyle{ \oplus }$\oplus
xor
propositional logic, Boolean algebra

T

1
Tautology The statement ⊤ is unconditionally true. A ⇒ ⊤ is always true. U+22A4 T $\displaystyle{ \top }$\top
top
propositional logic, Boolean algebra

F

0
Contradiction The statement ⊥ is unconditionally false. ⊥ ⇒ A is always true. U+22A5 &perp;
F
$\displaystyle{ \bot }$\bot
bottom
propositional logic, Boolean algebra
universal quantification ∀ x: P(x) means P(x) is true for all x. ∀ n ∈ N: n2 ≥ n. U+2200 &forall; $\displaystyle{ \forall }$\forall
for all; for any; for each
predicate logic
existential quantification ∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ N: n is even. U+2203 &exist; $\displaystyle{ \exists }$\exists
there exists
first-order logic
∃!
uniqueness quantification ∃! x: P(x) means there is exactly one x such that P(x) is true. ∃! n ∈ N: n + 5 = 2n. U+2203 U+0021 &exist; ! $\displaystyle{ \exists ! }$\exists !
there exists exactly one
first-order logic
:=

:⇔
definition x := y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence).

P :⇔ Q means P is defined to be logically equivalent to Q.
cosh x := (1/2)(exp x + exp (−x))

A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B)
U+003A U+003D

U+2261

U+003A U+229C
:=
: &equiv;
&hArr;
$\displaystyle{ := }$:=
$\displaystyle{ \equiv }$\equiv
$\displaystyle{ \Leftrightarrow }$\Leftrightarrow
is defined as
everywhere
( )
precedence grouping Perform the operations inside the parentheses first. (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4. U+0028 U+0029 ( ) $\displaystyle{ (~) }$ ( )
everywhere
inference x Template:Unicode y means y is derived from x. AB Template:Unicode ¬B → ¬A U+22A2 $\displaystyle{ \vdash }$\vdash
infers or is derived from
propositional logic, first-order logic

## Set Theory

* $\displaystyle{ \{a \in \mathbf A } }$  is used to denote that a is an element of a set  A.

• $\displaystyle{ \{x \in \mathbf R: x = x^2 \} \,\! }$ is the set $\displaystyle{ \{0, 1\} }$,
• $\displaystyle{ \{x \in \mathbf R: x \gt 0\} }$ is the set of all positive real numbers.

Cardinal Number

In formal set theory, a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method of counting sets using it gives the same result. (This is not true for the ordinal numbers.) In fact, the cardinal numbers are obtained by collecting all ordinal numbers which are obtainable by counting a given set.

Function

A function is a relation that uniquely associates members of one set with members of another set. More formally, a function from to is an object such that every is uniquely associated with an object . A function is therefore a many-to-one (or sometimes one-to-one) relation. The set of values at which a function is defined is called its domain, while the set of values that the function can produce is called its range. The term "map" is synonymous with function

## Basic Calc

Integral - An integral is a mathematical object that can be interpreted as an area or a generalization of area. Integrals, together with derivatives, are the fundamental objects of calculus. Other words for integral include antiderivative and primitive

Derivative - The derivative of a function represents an infinitesimal change in the function with respect to one of its variables.

The "simple" derivative of a function with respect to a variable is denoted either or

d f -- dx

1. Although this character is available in LaTeX, the Mediawiki TeX system doesn't support this character.