Statements - Revision history

Lila: /* Licensing */

2021-11-17T20:31:55Z

Licensing

Lila at 20:31, 17 November 2021

2021-11-17T20:31:18Z

Lila at 20:30, 17 November 2021

2021-11-17T20:30:09Z

Khanh at 20:36, 6 November 2021

2021-11-06T20:36:13Z

Lila at 19:03, 27 September 2021

2021-09-27T19:03:35Z

Lila at 22:05, 21 September 2021

2021-09-21T22:05:45Z

Lila: /* Connectives */

2021-09-21T22:04:01Z

Connectives

Lila at 21:15, 21 September 2021

2021-09-21T21:15:22Z

Lila: /* Connectives */

2021-09-21T21:02:35Z

Connectives

Lila: /* Connectives */

2021-09-21T21:00:25Z

Connectives

← Older revision		Revision as of 20:31, 17 November 2021
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	Content obtained and/or adapted from:		Content obtained and/or adapted from:
	*[https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/2%3A_Logical_Reasoning/2.1%3A_Statements_and_Logical_Operators Statements and Logical Operators, OpenStax] under a CC BY-NC-SA license		*[https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/2%3A_Logical_Reasoning/2.1%3A_Statements_and_Logical_Operators Statements and Logical Operators, OpenStax] under a CC BY-NC-SA license
		+	*[https://en.wikipedia.org/wiki/Theorem Theorem, Wikipedia] under a CC BY-NC-SA license

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 * A ''proposition'' is a theorem of lesser importance, or one that is considered so elementary or immediately obvious, that it may be stated without proof. This should not be confused with "proposition" as used in propositional logic. In classical geometry the term "proposition" was used differently: in Euclid's ''Elements'', all theorems and geometric constructions were called "propositions" regardless of their importance.
 * A ''lemma'' is an "accessory proposition" - a proposition with little applicability outside its use in a particular proof. Over time a lemma may gain in importance and be considered a ''theorem'', though the term "lemma" is usually kept as part of its name (e.g. Gauss's lemma, Zorn's lemma, and the fundamental lemma).
-* A ''corollary'' is a proposition that follows immediately from another theorem or axiom, with little or no required proof.<ref>Wentworth & Smith, article 51</ref> A corollary may also be a restatement of a theorem in a simpler form, or for a special case: for example, the theorem "all internal angles in a rectangle are right angles" has a corollary that "all internal angles in a ''square'' are right angles" - a square being a special case of a rectangle.
+* A ''corollary'' is a proposition that follows immediately from another theorem or axiom, with little or no required proof. A corollary may also be a restatement of a theorem in a simpler form, or for a special case: for example, the theorem "all internal angles in a rectangle are right angles" has a corollary that "all internal angles in a ''square'' are right angles" - a square being a special case of a rectangle.
 * A ''generalization'' of a theorem is a theorem with a similar statement but a broader scope, from which the original theorem can be deduced as a special case (a ''corollary'').

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 * All multiples of <math> 3 </math> are even (FALSE)
-===Connectives===
+==Terminology==
 We can use connectives to create compound statements out of two or more basic statements, or to transform statements. Here are some common connectives and how they are used (let <math> P </math> and <math> Q </math> be statements):
 * Conjunction: "<math> P </math> and <math> Q </math>", also denoted as "<math> P \and Q </math>". A conjunctive statement is only true if both <math> P </math> and <math> Q </math> are true, and false otherwise.

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 ==Resources==
 * [https://link-springer-com.libweb.lib.utsa.edu/content/pdf/10.1007%2F978-1-4419-7127-2.pdf Course Textbook], pages 4-25
-* [https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/2%3A_Logical_Reasoning/2.1%3A_Statements_and_Logical_Operators Statements and Logical Operators], OpenStax

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 In mathematics, a statement (also known as a proposition) is a declarative sentence that is either true or false but not both. There must be no ambiguity as to whether the statement is true or false. The statement cannot be an opinion or vague sentence (for example, "chocolate tastes good" or "red is an ugly color") that does not have a set truth value. Some examples of basic statements and their truth values:
-* 2 is an even number (TRUE)
+* <math> 2 </math> is an even number (TRUE)
-* Negative numbers are less than 0 (TRUE)
+* Negative numbers are less than <math> 0 </math> (TRUE)
-* 3 + 4 = 6 (FALSE)
+* <math> 3 + 4 = 6 </math> (FALSE)
 * Quadrilaterals have five sides (FALSE)
 * For all real numbers <math> n </math>, <math> n^2 \ge 0 </math> (TRUE)
-* All multiples of 3 are even (FALSE)
+* All multiples of <math> 3 </math> are even (FALSE)
 ===Connectives===
-We can use connectives to create compound statements out of two or more basic statements, or to transform statements. Here are some common connectives and how they are used (let P and Q be statements):
+We can use connectives to create compound statements out of two or more basic statements, or to transform statements. Here are some common connectives and how they are used (let <math> P </math> and <math> Q </math> be statements):
-* Conjunction: "P and Q", also denoted as "P ∧ Q". A conjunctive statement is only true if both P and Q are true, and false otherwise.
+* Conjunction: "<math> P </math> and <math> Q </math>", also denoted as "<math> P \and Q </math>". A conjunctive statement is only true if both <math> P </math> and <math> Q </math> are true, and false otherwise.
-* Disjunction: " P or Q", also denoted as "P ∨ Q". A disjunctive statement is true if either P or Q are true, and false only when both P and Q are false.
+* Disjunction: "<math> P </math> or <math> Q </math>", also denoted as "<math> P \or Q </math>". A disjunctive statement is true if either <math> P </math> or <math> Q </math> are true, and false only when both <math> P </math> and <math> Q </math> are false.
-* Negation: "not P", also denoted as "¬P". A negation reverses the truth value of the original statement (that is, if P is true then ¬P is false, and vice versa).
+* Negation: "not <math> P </math>", also denoted as "<math> \neg P </math>". A negation reverses the truth value of the original statement (that is, if <math> P </math> is true then <math> \neg P </math> is false, and vice versa).
-* Implication: "if P, then Q" or "P implies Q", also denoted as "P <math> \Rightarrow </math> Q". An implication is true if P and Q are both true, and false if P is true and Q is false. An implication is true when P is false, regardless of the truth value of Q (that is, Q can be true or false when P is false).
+* Implication: "if <math> P </math>, then <math> Q </math>" or "<math> P </math> implies <math> Q </math>", also denoted as "<math> P \rightarrow Q</math>". An implication is true if <math> P </math> and <math> Q </math> are both true, as well as when <math> P </math> is false, regardless of the truth value of <math> Q </math> (that is, <math> Q </math> can be true or false when <math> P </math> is false, and the implication will still be true). An implication is only false if <math> P </math> is true and <math> Q </math> is false.
-* Biconditional: "P if and only if Q", also denoted as "P <math> \Leftrightarrow </math> Q". A biconditional is only true when both "P <math> \Rightarrow </math> Q" and "Q <math> \Rightarrow </math> P" are true statements; that is, "P <math> \Leftrightarrow </math> Q" is true when P and Q have the same truth value, and false when they have different truth values.
+* Biconditional: "<math> P </math> if and only if <math> Q </math>", also denoted as "<math> P \Leftrightarrow Q </math>". A biconditional is only true when both <math> P \Rightarrow Q </math> and <math> Q \Rightarrow P </math>" are true statements; that is, <math> P \Leftrightarrow Q </math> is true when <math> P </math> and <math> Q </math> have the same truth value, and false when <math> P </math> and <math> Q </math> have different truth values.
-Truth table of some basic logical statements given statements P and Q:
+Truth table of some basic logical statements given statements <math> P </math> and <math> Q </math>:
 {| class="wikitable" style="margin:1em auto 1em auto; text-align:center;"
 |-

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 {| class="wikitable" style="margin:1em auto 1em auto; text-align:center;"
 |-
-! <math>P</math> || <math>Q</math> || <math>P \land Q</math> || <math>P \lor Q</math> || <math>P \Rightarrow Q</math> || <math>Q \Rightarrow P</math>  ||  <math>P \Leftrightarrow Q</math>
+! <math>P</math> || <math>Q</math> || <math>P \land Q</math> || <math>P \lor Q</math> || <math>P \Rightarrow Q</math> || <math>Q \Rightarrow P</math>  ||  <math>P \Leftrightarrow Q</math> || <math> \neg P </math> || <math> \neg P </math>
 |-
-| T || T || T || T || T || T || T
+| T || T || T || T || T || T || T || F || F
 |-
-| T || F || F || T || F || T || F
+| T || F || F || T || F || T || F || F || T
 |-
-| F || T || F || T || T || F || F
+| F || T || F || T || T || F || F || T || F
 |-
-| F || F || F || F || T || T || T
+| F || F || F || F || T || T || T || T || T
 |}