Concept: Truth Tables

Truth tables are an easy way to define what a logical expression means. Basically they are a table where each column is an expression, and each row indicates if that expression is true (T) or false (F).

For example, I want to define a concept I'll call sunrain, as when it is raining and it is sunny. I'll construct a truth table for sunrain so that there is no doubt in what I mean.

it is sunnyit is rainingit is sunrain
FFF
FTF
TFF
TTT

As you can see this table doesn't tell me if it is really sunraining right now, but it defines in every possible situation, whether that situation should be called sunrain. So if I look at row 2, I see that if it is not sunny, though it is raining, it's false to say that it is sunrain.

Truth tables can help simpify a somewhat complex idea. In a more abstract sense, I can define X as true when "A AND B are true, or when B is not true, but C is true". So X = (A AND B) OR (NOT B AND C).

 A  B  C A AND BNOT BNOT B AND C(A AND B)
OR
(NOT B AND C)
FFFFTFF
FFTFTTT
FTFFFFF
FTTFFFF
TFFFTFF
TFTFTTT
TTFTFFT
TTTTFFT

That's somewhat of a confusing definition, but just from a glance at the table we can see that X is true in four cases, and false in the other four. This table also helps in breaking down the definition, where each column builds on the previous columns, until the last one has the complete definition.

Below is a truth table of some basic expressions (which we'll talk about on later pages).

                        
FFFTFTFTTTTF
FTTFFTTTFTFT
TFTFFTTFTTFT
TTTFTFFFFFFF

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