Set-theory and logical statements generally have their own notation. While these topics do not properly belong within the subject of algebra, their notation often arises in algebra courses.

Oddly, though logic was studied before algebra was invented, the notation for logical statements (and the sets in whose context those statements are sometimes made) is less standardized than is the notation for algebra. Don't be surprised if your book sometimes uses notation that differs from what I've displayed below.

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Type-set formatting |
Text-only formatting |
Notes |

(2, 3) | (2, 3) | Put points in parentheses. Square brackets and other notation (or nothing at all) have other meanings. |

(2, 3) | (2, 3) | When you are writing an open interval, use parentheses, and note that "this is an interval", to differentiate an interval from a point. |

[2, 3] | [2, 3] | Use square brackets to indicate closed intervals. |

[2, ∞) | [2, inf.) [2, infinity) |
For "infinity", either spell out the word or else abbreviate as "inf.". Do not try to approximate the infinity symbol with two lower-case O's. Never use a square bracket at infinity, by the way. "Infinity" is not a number, so you can't "include" it in the interval. |

A ∪ B | A-union-B A U B |
If you don't spell out the set union (as in the first line to the left), then define your notation. |

A ∩ B |
A-intersect-B A ^ B |
If you don't spell out the set intersection, define your notation, so the reader doesn't mistake the carat to be indicating an exponent. |

A ⊂ B | A-subset-B A < B |
If you don't spell out the subset relation, define what you mean by the "less than" symbol. |

A ⊆ B | A-subset or equal-B A <= B |
If you don't spell out the subset relation, define your "less than or equal to" notation. |

A ⊄ B | A-not a subset-B | Spell out this relation. |

A-complement A^c |
Either spell out the complement relation, or define "to the power c" as being "the set complement". | |

A – B | A-complement-B A - B |
You can spell out the complement relation if you like, but pretty much everybody understands the "minus" notation. |

a ∈ A | a is in A a-element of-A |
The "is an element of" symbol is a tough one. Do not try to approximate it using a capital E; even though "everybody" tries to use it, readers are almost uniformly confused by it. (You'd think we'd learn, but... no.) Just spell out the relationship. |

a ∉ A | a is not in A a-not an element of-A |
Spell out the relation. |

∅ | {} the empty set Ø |
Curly braces are commonly used to denote sets, so you can use curly braces with nothing between to denote the empty set. Or (on a PC) hold down the "ALT" key and type "0216" on the numeric keypad. |

{1, 2, 3} | {1, 2, 3} | Sets are customarily written using curly braces. Don't use parentheses or other grouping symbols or, worse yet, no grouping symbols at all. |

N {1, 2, 3,...} the natural numbers |
If you use just the letter "N" to indicate "the set of natural numbers", tell the reader what you mean. | |

Z {..., -1, 0, 1, 2,...} the integers |
If you use just the letter "Z" to stand for "the set of integers, tell the reader what you mean. Otherwise, spell out the set you mean. | |

R the real numbers |
If you use just the letter "R" for "the set of all real numbers", define the notation. Otherwise, spell out the set you mean. | |

Q the rationals |
If you use just the letter "Q" for "the set of all rational numbers" (that is, the set of all fractions), define the notation. Otherwise, spell out the set you mean. | |

C the complex numbers |
If you use "C" for "the set of all complex numbers", define the notation. Otherwise, spell out the set you mean. | |

⇒ | if (this) then (that) ==> |
Either spell out the "if-then" relation, or approximate the arrow using two "equals" signs, to differentiate "if-then" from "greater than or equal to". |

⇔ | iff if and only if <==> |
The abbreviation "iff" for "if and only if" is fairly standard and should be recognized in context, but spell things out, if you're not sure. |

∃ | there exist(s) | Don't try to fake this symbol. Just spell out your meaning. |

∀ | for all | Don't try to fake this symbol. Just spell out your meaning. |

∧ | ^ and |
In the context of a logical statement, the carat should be recognized as meaning "and", but spell things out if you're not sure. |

∨ | v or |
Using "v" for the "or" symbol could be confusing, so either spell out your meaning or clearly define what you mean by "v". |

˜ ¬ |
~ ! not |
There are various symbols for "not", so define your notation, no matter which you use. In practice, it can often be simplest just to use "not-(whatever)". |

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