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Propositional formula
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=== IF ... THEN ... ELSE === This connective together with { 0, 1 }, ( or { F, T } or { <math>\bot</math>, <math>\top</math> } ) forms a complete set. In the following the IF...THEN...ELSE [[Relation (mathematics)|relation]] (c, b, a) = d represents ( (c β b) ∨ (~c β a) ) β‘ ( (c & b) ∨ (~c & a) ) = d : (c, b, a): : (c, 0, 1) β‘ ~c : (c, b, 1) β‘ (c β b) : (c, c, a) β‘ (c ∨ a) : (c, b, c) β‘ (c & b) Example: The following shows how a theorem-based proof of "(c, b, 1) β‘ (c β b)" would proceed, below the proof is its truth-table verification. ( Note: (c β b) is ''defined'' to be (~c ∨ b) ): :* Begin with the reduced form: ( (c & b) ∨ (~c & a) ) :* Substitute "1" for a: ( (c & b) ∨ (~c & 1) ) :* Identity (~c & 1) = ~c: ( (c & b) ∨ (~c) ) :* Law of commutation for V: ( (~c) ∨ (c & b) ) :* Distribute "~c V" over (c & b): ( ((~c) ∨ c ) & ((~c) ∨ b ) :* Law of excluded middle (((~c) ∨ c ) = 1 ): ( (1) & ((~c) ∨ b ) ) :* Distribute "(1) &" over ((~c) ∨ b): ( ((1) & (~c)) ∨ ((1) & b )) ) :* Commutivity and Identity (( 1 & ~c) = (~c & 1) = ~c, and (( 1 & b) β‘ (b & 1) β‘ b: ( ~c ∨ b ) :* ( ~c ∨ b ) is defined as '''c β b''' Q. E. D. In the following truth table the column labelled "taut" for tautology evaluates logical equivalence (symbolized here by β‘) between the two columns labelled d. Because all four rows under "taut" are 1's, the equivalence indeed represents a tautology. {|style="margin-left: auto; margin-right: auto; border: none;" |- style="font-size:9pt; text-align:center" | width="27.75" Height="12" | | width="20.25" | | width="18.75" | | width="18.75" | | width="6.75" | | width="11.25" | | width="11.25" | | width="11.25" | | width="11.25" | | width="11.25" | | width="11.25" | | width="11.25" | |style="background-color:#FDE9D9" width="11.25" | d | width="11.25" | | width="11.25" | | width="11.25" | | width="11.25" | | width="11.25" | | width="11.25" | | width="11.25" | | width="11.25" | | width="12.75" | |style="background-color:#DDD9C3" width="19.5" | taut | width="11.25" | | width="11.25" | | width="11.25" | | width="11.25" | | width="11.25" | |style="background-color:#FDE9D9" width="11.25" | d | width="11.25" | | width="12.75" | | width="12.75" | |- style="font-size:9pt;font-weight:bold" align="center" |style="background-color:#F2F2F2" Height="12" | rows | c | b | a |style="background-color:#A5A5A5" | | ( | ( | ( | c |style="background-color:#DBE5F1" | & | b | ) |style="background-color:#FDE9D9" | V | ( |style="background-color:#EAF1DD" | ~ | ( | c | ) |style="background-color:#DBE5F1" | & | a | ) | ) |style="background-color:#DDD9C3" | β‘ | ( |style="background-color:#EAF1DD" | ~ | ( | c | ) |style="background-color:#FDE9D9" | V | b | ) | ) |- style="font-size:9pt" align="center" |style="background-color:#F2F2F2" Height="12" | 0,1 | 0 | 0 | 1 |style="background-color:#A5A5A5" | | | | | 0 |style="background-color:#DBE5F1" | 0 | 0 | |style="background-color:#FDE9D9" | 1 | |style="background-color:#EAF1DD" | 1 | | 0 | |style="background-color:#DBE5F1" | 1 | 1 | | |style="background-color:#DDD9C3" | 1 | |style="background-color:#EAF1DD" | 1 | | 0 | |style="background-color:#FDE9D9" | 1 | 0 | | |- style="font-size:9pt" align="center" |style="background-color:#F2F2F2" Height="12" | 2,3 | 0 | 1 | 1 |style="background-color:#A5A5A5" | | | | | 0 |style="background-color:#DBE5F1" | 0 | 1 | |style="background-color:#FDE9D9" | 1 | |style="background-color:#EAF1DD" | 1 | | 0 | |style="background-color:#DBE5F1" | 1 | 1 | | |style="background-color:#DDD9C3" | 1 | |style="background-color:#EAF1DD" | 1 | | 0 | |style="background-color:#FDE9D9" | 1 | 1 | | |- style="font-size:9pt" align="center" |style="background-color:#F2F2F2" Height="12" | 4,5 | 1 | 0 | 1 |style="background-color:#A5A5A5" | | | | | 1 |style="background-color:#DBE5F1" | 0 | 0 | |style="background-color:#FDE9D9" | 0 | |style="background-color:#EAF1DD" | 0 | | 1 | |style="background-color:#DBE5F1" | 0 | 1 | | |style="background-color:#DDD9C3" | 1 | |style="background-color:#EAF1DD" | 0 | | 1 | |style="background-color:#FDE9D9" | 0 | 0 | | |- style="font-size:9pt" align="center" |style="background-color:#F2F2F2" Height="12" | 6,7 | 1 | 1 | 1 |style="background-color:#A5A5A5" | | | | | 1 |style="background-color:#DBE5F1" | 1 | 1 | |style="background-color:#FDE9D9" | 1 | |style="background-color:#EAF1DD" | 0 | | 1 | |style="background-color:#DBE5F1" | 0 | 1 | | |style="background-color:#DDD9C3" | 1 | |style="background-color:#EAF1DD" | 0 | | 1 | |style="background-color:#FDE9D9" | 1 | 1 | | |}
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