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MyWikiBiz, Author Your Legacy — Wednesday April 09, 2025
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A '''truth table''' is a tabular array that illustrates the computation of a [[boolean function]], that is, a function of the form ''f'' : '''B'''<sup>''k''</sup> → '''B''', where ''k'' is a non-negative integer and '''B''' is the [[boolean domain]] {0, 1}.
+A '''truth table''' is a tabular array that illustrates the computation of a [[boolean function]], that is, a function of the form <math>f : \mathbb{B}^k \to \mathbb{B},</math> where <math>k\!</math> is a non-negative integer and <math>\mathbb{B}</math> is the [[boolean domain]] <math>\{ 0, 1 \}.\!</math>
==Logical negation==
==Logical negation==
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The truth table of '''NOT p''' (also written as '''~p''' or '''¬p''') is as follows:
The truth table of '''NOT p''' (also written as '''~p''' or '''¬p''') is as follows:
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:40%"
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:40%"
|+ '''Logical Negation'''
|+ '''Logical Negation'''
−|- style="background:paleturquoise"
+|- style="background:#e6e6ff"
! style="width:20%" | p
! style="width:20%" | p
! style="width:20%" | ¬p
! style="width:20%" | ¬p
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| T || F
| T || F
|}
|}
+
<br>
<br>
The logical negation of a proposition '''p''' is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following:
The logical negation of a proposition '''p''' is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following:
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; width:40%"
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; width:40%"
|+ '''Variant Notations'''
|+ '''Variant Notations'''
−|- style="background:paleturquoise"
+|- style="background:#e6e6ff"
! style="text-align:center" | Notation
! style="text-align:center" | Notation
! Vocalization
! Vocalization
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| bang ''p''
| bang ''p''
|}
|}
+
<br>
<br>
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The truth table of '''p AND q''' (also written as '''p ∧ q''', '''p & q''', or '''p<math>\cdot</math>q''') is as follows:
The truth table of '''p AND q''' (also written as '''p ∧ q''', '''p & q''', or '''p<math>\cdot</math>q''') is as follows:
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Conjunction'''
|+ '''Logical Conjunction'''
−|- style="background:paleturquoise"
+|- style="background:#e6e6ff"
! style="width:15%" | p
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | q
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| T || T || T
| T || T || T
|}
|}
+
<br>
<br>
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The truth table of '''p OR q''' (also written as '''p ∨ q''') is as follows:
The truth table of '''p OR q''' (also written as '''p ∨ q''') is as follows:
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Disjunction'''
|+ '''Logical Disjunction'''
−|- style="background:paleturquoise"
+|- style="background:#e6e6ff"
! style="width:15%" | p
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | q
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| T || T || T
| T || T || T
|}
|}
+
<br>
<br>
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The truth table of '''p EQ q''' (also written as '''p = q''', '''p ↔ q''', or '''p ≡ q''') is as follows:
The truth table of '''p EQ q''' (also written as '''p = q''', '''p ↔ q''', or '''p ≡ q''') is as follows:
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Equality'''
|+ '''Logical Equality'''
−|- style="background:paleturquoise"
+|- style="background:#e6e6ff"
! style="width:15%" | p
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | q
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| T || T || T
| T || T || T
|}
|}
+
<br>
<br>
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The truth table of '''p XOR q''' (also written as '''p + q''', '''p ⊕ q''', or '''p ≠ q''') is as follows:
The truth table of '''p XOR q''' (also written as '''p + q''', '''p ⊕ q''', or '''p ≠ q''') is as follows:
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
|+ '''Exclusive Disjunction'''
|+ '''Exclusive Disjunction'''
−|- style="background:paleturquoise"
+|- style="background:#e6e6ff"
! style="width:15%" | p
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | q
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| T || T || F
| T || T || F
|}
|}
+
<br>
<br>
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The truth table associated with the material conditional '''if p then q''' (symbolized as '''p → q''') and the logical implication '''p implies q''' (symbolized as '''p ⇒ q''') is as follows:
The truth table associated with the material conditional '''if p then q''' (symbolized as '''p → q''') and the logical implication '''p implies q''' (symbolized as '''p ⇒ q''') is as follows:
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Implication'''
|+ '''Logical Implication'''
−|- style="background:paleturquoise"
+|- style="background:#e6e6ff"
! style="width:15%" | p
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | q
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| T || T || T
| T || T || T
|}
|}
+
<br>
<br>
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The truth table of '''p NAND q''' (also written as '''p | q''' or '''p ↑ q''') is as follows:
The truth table of '''p NAND q''' (also written as '''p | q''' or '''p ↑ q''') is as follows:
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
|+ '''Logical NAND'''
|+ '''Logical NAND'''
−|- style="background:paleturquoise"
+|- style="background:#e6e6ff"
! style="width:15%" | p
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | q
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| T || T || F
| T || T || F
|}
|}
+
<br>
<br>
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The truth table of '''p NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows:
The truth table of '''p NNOR q''' (also written as '''p ⊥ q''' or '''p ↓ q''') is as follows:
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
+<br>
+{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:45%"
|+ '''Logical NNOR'''
|+ '''Logical NNOR'''
−|- style="background:paleturquoise"
+|- style="background:#e6e6ff"
! style="width:15%" | p
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | q
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| T || T || F
| T || T || F
|}
|}
+
<br>
<br>
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[[Category:Computer Science]]
[[Category:Computer Science]]
[[Category:Discrete Mathematics]]
[[Category:Discrete Mathematics]]
+[[Category:Formal Languages]]
+[[Category:Formal Sciences]]
+[[Category:Formal Systems]]
+[[Category:Linguistics]]
[[Category:Logic]]
[[Category:Logic]]
[[Category:Mathematics]]
[[Category:Mathematics]]
+[[Category:Philosophy]]
+[[Category:Semiotics]]
+
+<sharethis />
