## Truth Table

• truth table

### Contents

Often we want to discuss properties/relations common to all propositions. In such a case rather than stating them for each individual proposition we use variables representing an arbitrary proposition and state properties/relations in terms of those variables. Those variables are called a propositional variable. Propositional variables are also considered a proposition and called a proposition since they represent a proposition hence they behave the same way as propositions. A proposition in general contains a number of variables. For example (P Q) contains variables P and Q each of which represents an arbitrary proposition. Thus a proposition takes different values depending on the values of the constituent variables. This relationship of the value of a proposition and those of its constituent variables can be represented by a table. It tabulates the value of a proposition for all possible values of its variables and it is called a truth table.

For example the following table shows the relationship between the values of P, Q and P Q:

OR
P Q (P Q)
F F F
F T T
T F T
T T T

In the table, F represents truth value false and T true.
This table shows that P Q is false if P and Q are both false, and it is true in all the other cases.

### Test Your Understanding of Truth Table

Which of the following tables are a truth table ?
Z below represents a proposition involving P and Q.

Table 1
P Q Proposition Z
F F F
T F T
T T T
T F T
Table 2
P Q Proposition Z
F F F
T F T
T T F

Table 3
P Q Proposition Z
F F F
F T T
T F T
T T T
Table 4
P Proposition Z
F F
F T
T F

For each of the above tables, click "Yes" if it is a truth table, else click "No" below, then click Submit.

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