Editing Disjunctive normal form (section)

=== ... by semantic means ===
The full DNF of a formula can be read off its [[truth table]].<ref>{{harvnb|Smullyan|1968|p=14}}: "Make a truth-table for the formula. Each line of the table which comes out "T" will yield one of the basic conjunctions of the disjunctive normal form."</ref>{{sfn|Sobolev|2020}} For example, consider the formula
:<math>\phi = ((\lnot (p \land q)) \leftrightarrow (\lnot r \uparrow (p \oplus q)))</math>.<ref><math>\phi</math> = (('''[[Negation|NOT]]''' (p '''[[Logical conjunction|AND]]''' q)) '''[[If and only if|IFF]]''' (('''[[Negation|NOT]]''' r) '''[[Sheffer stroke|NAND]]''' (p '''[[XOR]]''' q)))</ref>

The corresponding [[truth table]] is
:{| class="wikitable" style="text-align:center;"
! <math>p</math>
! <math>q</math>
! <math>r</math>
! style="background:black"| 
! <math>(</math>
! <math>\lnot</math>
! <math>(p \land q)</math>
! <math>)</math>
! <math>\leftrightarrow</math>
! <math>(</math>
! <math>\lnot r</math>
! <math>\uparrow</math>
! <math>(p \oplus q)</math>
! <math>)</math>
|-
| T || T || T ||style="background:black"| || || F || T || || style="background:papayawhip" | F || || F || T || F ||
|-
| T || T || F ||style="background:black"| || || F || T || || style="background:papayawhip" | F || || T || T || F ||
|-
| style="background:lightgreen"|T || style="background:lightgreen"|F || style="background:lightgreen"|T ||style="background:black"| || || T || F || || style="background:papayawhip" | '''T''' || || F || T || T ||
|-
| T || F || F ||style="background:black"| || || T || F || || style="background:papayawhip" | F || || T || F || T ||
|-
| style="background:lightgreen"|F || style="background:lightgreen"|T || style="background:lightgreen"|T ||style="background:black"| || || T || F || || style="background:papayawhip" | '''T''' || || F || T || T ||
|-
| F || T || F ||style="background:black"| || || T || F || || style="background:papayawhip" | F || || T || F || T ||
|-
| style="background:lightgreen"|F || style="background:lightgreen"|F || style="background:lightgreen"|T ||style="background:black"| || || T || F || || style="background:papayawhip" | '''T''' || || F || T || F ||
|-
| style="background:lightgreen"|F || style="background:lightgreen"|F || style="background:lightgreen"|F ||style="background:black"| || || T || F || || style="background:papayawhip" | '''T''' || || T || T || F ||
|}

* The full DNF equivalent of <math>\phi</math> is
:<math>
(      p \land \lnot q \land       r) \lor
(\lnot p \land       q \land       r) \lor
(\lnot p \land \lnot q \land       r) \lor
(\lnot p \land \lnot q \land \lnot r)
</math>
* The full DNF equivalent of <math>\lnot \phi</math> is
:<math>
(      p \land       q \land       r) \lor
(      p \land       q \land \lnot r) \lor
(      p \land \lnot q \land \lnot r) \lor
(\lnot p \land       q \land \lnot r)
</math>