Logic: Difference between revisions
No edit summary |
|||
| Line 9: | Line 9: | ||
A '''Logical notation''' is a special syntax that is shorthand for logical statements. | A '''Logical notation''' is a special syntax that is shorthand for logical statements. | ||
==Negations== | ===Negations=== | ||
A negation is denoted by <math>\neg p</math>. <math>\neg p</math> is the statement that is true when <math>p</math> is false and the statement that is false when <math>p</math> is true. This means simply "the opposite of <math>p</math>" | A negation is denoted by <math>\neg p</math>. <math>\neg p</math> is the statement that is true when <math>p</math> is false and the statement that is false when <math>p</math> is true. This means simply "the opposite of <math>p</math>" | ||
==Conjunction== | ===Conjunction=== | ||
The conjunction of two statements basically means "<math>p</math> and <math>q</math>" and is denoted by <math>p \land q</math>. | The conjunction of two statements basically means "<math>p</math> and <math>q</math>" and is denoted by <math>p \land q</math>. | ||
==Disjunction== | ===Disjunction=== | ||
The disjunction of two statements basically means "<math>p</math> or <math>q</math>" and is denoted by <math>p \land q</math>. | The disjunction of two statements basically means "<math>p</math> or <math>q</math>" and is denoted by <math>p \land q</math>. | ||
==Implication== | ===Implication=== | ||
This operation is given by the statement "If <math>p</math>, then <math>q</math>". It is denoted by <math>p\Leftrightarrow q</math>. An example is "if <math>x+3=5</math>, then <math>x=2</math>. | This operation is given by the statement "If <math>p</math>, then <math>q</math>". It is denoted by <math>p\Leftrightarrow q</math>. An example is "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 \Leftrightarrow q</math> is <math>q \Leftrightarrow p</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 \Leftrightarrow q</math> is <math>\neg p \Leftrightarrow \neg q</math> | ||
Revision as of 17:23, 22 November 2011
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 statement can be "A duck is a bird." which is true. Another example is "A pencil does not exist" which is false.
Logical Notations
- Main article: Logical notation
A Logical notation is a special syntax that is shorthand for logical statements.
Negations
A negation is denoted by 
. is the statement that is true when 
is false and the statement that is false when is true. This means simply "the opposite of "
Conjunction
The conjunction of two statements basically means " and 
" and is denoted by 
.
Disjunction
The disjunction of two statements basically means " or " and is denoted by .
Implication
This operation is given by the statement "If , then ". It is denoted by 
. An example is "if 
, then 
.
Converse
The converse of the statement 
is 
.
Contrapositive
The contrapositive of the statement is 
Truth Tables
A truth tale 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 
and an existential quantifier is denoted by 
.
