First Order Logic Resources
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Tutorials
 Rules and terminology by Wolfram Mathworld [FOL]
 Some [good note]s on how to start with FOL (very programming oriented)
 Peter Suber's [translating logic into English AND his [glossary] of set theory is immensely helpful for the uninitiated.
 [PD Magnus] forall x: an introduction to formal logic.
 Thorough intro, but very "textbookish" below
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Symbols in Logic
This chart can be found on the [wikipedia page]
Other Logic Resources
Other Tutorials
Symbol

Name  Explanation  Examples  Unicode Value 
HTML Entity 
LaTeX symbol 

Should be read as  
Category  
⇒
→ ⊃ 
material implication  A ⇒ B 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 ⇒ x^{2} = 4 is true, but x^{2} = 4 ⇒ x = 2 is in general false (since x could be −2).  U+21D2 U+2192 U+2283 
⇒ → ⊃ 
[math]\displaystyle{ \Rightarrow }[/math]\Rightarrow
[math]\displaystyle{ \to }[/math]\to [math]\displaystyle{ \supset }[/math]\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 
⇔ ≡ ↔ 
[math]\displaystyle{ \Leftrightarrow }[/math]\Leftrightarrow
[math]\displaystyle{ \equiv }[/math]\equiv [math]\displaystyle{ \leftrightarrow }[/math]\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 
¬ ˜ ~ 
[math]\displaystyle{ \lnot }[/math]\lnot
[math]\displaystyle{ \sim }[/math]\sim 
not  
propositional logic  
∧
• & 
logical conjunction  The statement A ∧ B 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 
∧ & 
[math]\displaystyle{ \land }[/math]\land \&^{[1]} 
and  
propositional logic  
∨
+ 
logical disjunction  The statement A ∨ B 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  ∨  [math]\displaystyle{ \lor }[/math]\lor 
or  
propositional logic  
exclusive disjunction  The statement A ⊕ B is true when either A or B, but not both, are true. A Template:Unicode B means the same.  (¬A) ⊕ A is always true, A ⊕ A is always false.  U+2295 U+22BB 
⊕  [math]\displaystyle{ \oplus }[/math]\oplus  
xor  
propositional logic, Boolean algebra  
⊤ T 1 
Tautology  The statement ⊤ is unconditionally true.  A ⇒ ⊤ is always true.  U+22A4  T  [math]\displaystyle{ \top }[/math]\top 
top  
propositional logic, Boolean algebra  
⊥ F 0 
Contradiction  The statement ⊥ is unconditionally false.  ⊥ ⇒ A is always true.  U+22A5  ⊥ F 
[math]\displaystyle{ \bot }[/math]\bot 
bottom  
propositional logic, Boolean algebra  
∀

universal quantification  ∀ x: P(x) means P(x) is true for all x.  ∀ n ∈ N: n^{2} ≥ n.  U+2200  ∀  [math]\displaystyle{ \forall }[/math]\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  ∃  [math]\displaystyle{ \exists }[/math]\exists 
there exists  
firstorder 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  ∃ !  [math]\displaystyle{ \exists ! }[/math]\exists ! 
there exists exactly one  
firstorder 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 
:= : ≡ ⇔ 
[math]\displaystyle{ := }[/math]:=
[math]\displaystyle{ \equiv }[/math]\equiv [math]\displaystyle{ \Leftrightarrow }[/math]\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  ( )  [math]\displaystyle{ (~) }[/math] ( ) 
everywhere  
inference  x Template:Unicode y means y is derived from x.  A → B Template:Unicode ¬B → ¬A  U+22A2  [math]\displaystyle{ \vdash }[/math]\vdash  
infers or is derived from  
propositional logic, firstorder logic 
Logical Operator With Venn Explanations
[Wikipedia Page] is really best resource.
Set Theory
* [math]\displaystyle{ \{a \in \mathbf A } }[/math] is used to denote that a is an element of a set A.
 [math]\displaystyle{ \{x \in \mathbf R: x = x^2 \} \,\! }[/math] is the set [math]\displaystyle{ \{0, 1\} }[/math],
 [math]\displaystyle{ \{x \in \mathbf R: x \gt 0\} }[/math] 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 manytoone (or sometimes onetoone) 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