Difference between revisions of "Statements"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
Line 15: Line 15:
 
* Implication: "if P, then Q" or "P implies Q", also denoted as "P <math> \implies </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 P, then Q" or "P implies Q", also denoted as "P <math> \implies </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).
 
* Biconditional: "P if and only if Q", also denoted as "P <math> \iff </math> Q". A biconditional is only true when both "P <math> \implies </math> Q" and "Q <math> \implies </math> P" are true statements; that is, "P <math> \iff </math> Q" is true when P and Q have the same truth value, and false when they have different truth values.
 
* Biconditional: "P if and only if Q", also denoted as "P <math> \iff </math> Q". A biconditional is only true when both "P <math> \implies </math> Q" and "Q <math> \implies </math> P" are true statements; that is, "P <math> \iff </math> Q" is true when P and Q have the same truth value, and false when they have different truth values.
 +
 +
{| class="wikitable" style="margin:1em auto 1em auto; text-align:center;"
 +
|-
 +
! {{nobold|{{mvar|P}}}} || {{nobold|{{mvar|Q}}}} || <math>P \land Q</math> || <math>P \lor Q</math> || <math>P\ \underline{\lor}\ Q</math> || <math>P\ \underline{\land}\ Q</math> || <math>P \Rightarrow Q</math> || <math>P \Leftarrow Q</math>  ||  <math>P \Leftrightarrow Q</math>
 +
|-
 +
| T || T ||  T  ||  T  ||  style="background:papayawhip" | F  ||  T  ||  T  ||  T  ||  T
 +
|-
 +
| T || style="background:papayawhip" | F  ||  style="background:papayawhip" | F  ||  T  ||  T  ||  style="background:papayawhip" | F  ||  style="background:papayawhip" | F  ||  T  ||  style="background:papayawhip" | F
 +
|-
 +
| style="background:papayawhip" | F  || T ||  style="background:papayawhip" | F  ||  T  ||  T  ||  style="background:papayawhip" | F  ||  T  ||  style="background:papayawhip" | F  ||  style="background:papayawhip" | F
 +
|-
 +
| style="background:papayawhip" | F  || style="background:papayawhip" | F  ||  style="background:papayawhip" | F  ||  style="background:papayawhip" | F  ||  style="background:papayawhip" | F  ||  T  ||  T  ||  T  ||  T
 +
|-
 +
! {{nobold|{{mvar|P}}}} || {{nobold|{{mvar|Q}}}} || <math>P \land Q</math> || <math>P \lor Q</math> || <math>P\ \underline{\lor}\ Q</math> || <math>P\ \underline{\land}\ Q</math> || <math>P \Rightarrow Q</math> || <math>P \Leftarrow Q</math>  ||  <math>P \Leftrightarrow Q</math>
 +
|-
 +
| ||
 +
| [[logical conjunction|AND]]<br /> (conjunction)
 +
| [[logical disjunction|OR]] <br /> (disjunction)
 +
| [[Exclusive or|XOR]]      <br /> (exclusive or) <!-- this could be "+" instead according to other articles -->
 +
| [[Exclusive nor|XNOR]]    <br /> (exclusive nor)
 +
| [[logical conditional|conditional <br /> "if-then"]]
 +
| conditional <br /> "then-if"
 +
| [[if and only if|biconditional<br /> "if-and-only-if"]]
 +
|-
 +
| colspan=18 |
 +
''where'' &nbsp;{{colorbox|white|&nbsp;T&nbsp;}}&nbsp; ''means'' '''true''' ''and'' &nbsp;{{colorbox|papayawhip|&nbsp;F&nbsp;}}&nbsp; ''means'' '''false'''
 +
|}
  
 
==Resources==
 
==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://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
 
* [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

Revision as of 14:56, 21 September 2021

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)
  • Negative numbers are less than 0 (TRUE)
  • 3 + 4 = 6 (FALSE)
  • Quadrilaterals have five sides (FALSE)
  • For all real numbers , (TRUE)
  • All multiples of 3 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):

  • 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.
  • 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.
  • 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).
  • Implication: "if P, then Q" or "P implies Q", also denoted as "P 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).
  • Biconditional: "P if and only if Q", also denoted as "P Q". A biconditional is only true when both "P Q" and "Q P" are true statements; that is, "P Q" is true when P and Q have the same truth value, and false when they have different truth values.
Template:Nobold Template:Nobold
T T T T F T T T T
T F F T T F F T F
F T F T T F T F F
F F F F F T T T T
Template:Nobold Template:Nobold
AND
(conjunction)
OR
(disjunction)
XOR
(exclusive or)
XNOR
(exclusive nor)
conditional
"if-then"
conditional
"then-if"
biconditional
"if-and-only-if"

where  Template:Colorbox  means true and  Template:Colorbox  means false

Resources