Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Logic"

(→‎Disjunction)
m (→‎Implication, Conditional: wording)
 
(9 intermediate revisions by 4 users not shown)
Line 2: Line 2:
  
 
==Statements==
 
==Statements==
−
A statement is either true or false, but it will never be both or neither. An example of statement can be "A duck is a bird." which is true. Another example is "A pencil does not exist" which is false.
+
A statement is either true or false, but it will never be both or neither. An example of a statement is "A duck is a bird" which is true. Another example is "A pencil does not exist" which is false.
 +
 
 +
===Conditional===
 +
If <math>P</math> then <math>Q</math>. For example, "If it is a duck then it is a bird."
 +
 
 +
===Inverse===
 +
The inverse of the conditional statement is: If not <math>P</math> then not <math>Q</math>.
 +
 
 +
===Converse===
 +
The converse of the conditional statement is: If <math>Q</math> then <math>P</math>.
 +
 
 +
===Contrapositive===
 +
The contrapositive of the conditional statement is: If not <math>Q</math> then not <math>P</math>.
 +
 
 +
<br />
 +
The conditional is equivalent to the contrapositive. The inverse is equivalent to the converse. When both the conditional and the converse are true at the same time, this is equivalent to an [[Iff]] statement.
  
 
==Logical Notations==
 
==Logical Notations==
Line 18: Line 33:
 
The disjunction of two statements basically means "<math>p</math> or <math>q</math>" and is denoted by <math>p \vee q</math>.
 
The disjunction of two statements basically means "<math>p</math> or <math>q</math>" and is denoted by <math>p \vee q</math>.
  
−
===Implication===
+
===Implication, Conditional===
−
This operation is given by the statement "If <math>p</math>, then <math>q</math>". It is denoted by <math>p\implies q</math>. An example is "if <math>x+3=5</math>, then <math>x=2</math>.
+
The statement "If <math>p</math> then <math>q</math>" is denoted by <math>p\implies q</math>. For example, <math>x+3=5\implies x=2</math> means "If <math>x+3=5</math> then <math>x=2</math>."
  
 
===Converse===
 
===Converse===
−
The converse of the statement <math>p \Leftrightarrow q</math> is <math>q \Leftrightarrow p</math>.
+
The converse of the statement <math>p \implies q</math> is <math>q \implies p</math>.
 +
 
 +
===Inverse===
 +
The inverse of the statement <math>p \implies q</math> is <math>\neg p \implies \neg q</math>.
  
 
===Contrapositive===
 
===Contrapositive===
−
The contrapositive of the statement <math>p \Leftrightarrow q</math> is <math>\neg p \Leftrightarrow \neg q</math>
+
The contrapositive of the statement <math>p \implies q</math> is <math>\neg q \implies \neg p</math>. These statements are logically equivalent.
  
 
==Truth Tables==
 
==Truth Tables==
−
A truth tale is the list of all possible values of a compound statement.
+
A truth table is the list of all possible values of a compound statement.
  
 
==Quantifiers==
 
==Quantifiers==

Latest revision as of 16:22, 31 July 2020

Logic is the systematic use of symbolic and mathematical techniques to determine the forms of valid deductive or inductive argument.

Statements

A statement is either true or false, but it will never be both or neither. An example of a statement is "A duck is a bird" which is true. Another example is "A pencil does not exist" which is false.

Conditional

If $P$ then $Q$. For example, "If it is a duck then it is a bird."

Inverse

The inverse of the conditional statement is: If not $P$ then not $Q$.

Converse

The converse of the conditional statement is: If $Q$ then $P$.

Contrapositive

The contrapositive of the conditional statement is: If not $Q$ then not $P$.


The conditional is equivalent to the contrapositive. The inverse is equivalent to the converse. When both the conditional and the converse are true at the same time, this is equivalent to an Iff statement.

Logical Notations

Main article: Logical notation

A Logical notation is a special syntax that is shorthand for logical statements.

Negations

The negation of $p$, denoted by $\neg p$, is the statement that is true when $p$ is false and is false when $p$ is true. This means simply "it is not the case that $p$."

Conjunction

The conjunction of two statements basically means "$p$ and $q$" and is denoted by $p \land q$.

Disjunction

The disjunction of two statements basically means "$p$ or $q$" and is denoted by $p \vee q$.

Implication, Conditional

The statement "If $p$ then $q$" is denoted by $p\implies q$. For example, $x+3=5\implies x=2$ means "If $x+3=5$ then $x=2$."

Converse

The converse of the statement $p \implies q$ is $q \implies p$.

Inverse

The inverse of the statement $p \implies q$ is $\neg p \implies \neg q$.

Contrapositive

The contrapositive of the statement $p \implies q$ is $\neg q \implies \neg p$. These statements are logically equivalent.

Truth Tables

A truth table is the list of all possible values of a compound statement.

Quantifiers

There are two types of quantifiers: A universal Quantifier: "for all" and an existential Quantifier: "there exists". A universal quantifier is denoted by $\forall$ and an existential quantifier is denoted by $\exists$.

See Also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Logic&oldid=130038"