What is a truth table

Truth table - Serlo "Math for non-freaks"

In the following, we present the truth value table, or “truth table” for short. Such a table is helpful for examining statements of logic.

The truth table [edit]

Imagine you have a statement that is a combination of several atomic statements, , , ... is with junctions. The truth value of this composite statement can be determined unambiguously from the truth values ​​of these partial statements, because for each junctor it is specified how the truth value of the composite statement results from the partial statements. An example of this is the statement made up of the juncture “and” :

Accordingly, there is a clearly defined rule for a statement made up of several joiners, which determines what the truth value of this linked statement is as a function of its atomic statements. Therefore all possible assignments of the statements , , , ... and the associated resulting truth value of the entire statement are shown in a table. Such a table will be Truth table called. The following table clarifies the principle of truth tables:

A truth table is used to represent the truth value of a composite statement as a function of the truth values ​​of its atomic statements. The number of lines can quickly become large.

Comprehension question: How many lines are there in atomic statements necessary?

There are Lines necessary because for each of the atomic statements the two truth values ​​"true" and "false" are possible as an assignment. So with 2 partial statements 4, with 3 partial statements 8 and with 4 partial statements 16 lines are necessary.

So that you don't get confused with the possible combinations even with many atomic statements, it is a good strategy to orientate yourself on the binary system. The binary representation of a line number is converted into the truth values for 1 and implemented for 0:

Create truth tables [edit]

How many columns do we need in the truth table? Since the compound statement was built up step by step from the partial statements, you need a column for each of these partial statements. The first columns are the atomic statements, then the other partial statements follow and the last column then contains the entire statement.

Let's take the formula as an example

.

It has 3 atomic partial statements, namely , and . Hence the table Lines. Furthermore are partial statements , , . The first two form the partial statement , because binds stronger than . Finally, we have the full statement. Overall that results Partial statements, so Columns.