A truth table is a tabular representation of all possible combinations of truth values for a given logical expression. It is commonly used in logic and mathematics to analyze the behavior of logical statements. Here are the general steps to construct a truth table:
Identify the variables: Determine the variables in your logical expression. These are the symbols that can take on either a true (T) or false (F) value.
Determine the number of rows: The number of rows in the truth table is determined by the number of variables. One variable has two rows, true and false. For two variables, we have a total of four states. For three variables, the total number of states will be eight. The addition of each variable doubles the number of rows.
Number of variables | Number of rows |
1 | 2 = 2^{1} |
2 | 4 = 2^{2} |
3 | 8 = 2^{3} |
4 | 16 = 2^{4} |
Based on this, if we have $n$ variables, there will be $2^n$ rows in the table because each variable can take on two possible truth values (T or F).
List all combinations: Create a table with columns for each variable and a column for the final logical expression. List all possible combinations of truth values for the variables. Start with all variables being true (T) in the first row and alternate the truth values in subsequent rows.
Evaluate the logical expression: For each row, evaluate the logical expression based on the truth values of the variables. Use the logical operators AND (∧, &), OR ($∨,|$), NOT ($¬, !, '$), etc., to determine the truth value of the entire expression.
Complete the table: Fill in the column for the logical expression with the resulting truth values for each row.
Suppose we have a logical expression, $P∧Q∨¬R$.
We have a total of three variables, $P$, $Q$, and $R$.
Number of rows will be $2^3$ which will be $8$.
We have two combinations here, $P∧Q$ and $(P∧Q)∨¬R$.
Let’s fill in the column for the logical expression based on the truth values of $P$, $Q$, and $R$.
P | Q | R | P∧Q | ¬R | P∧Q∨¬R |
T | T | T | T | F | T |
T | T | F | T | T | T |
T | F | T | F | F | F |
T | F | F | F | T | T |
F | T | T | F | F | F |
F | T | F | F | T | T |
F | F | T | F | F | F |
F | F | F | F | T | T |
This process helps us systematically analyze the behavior of a logical expression and understand all possible outcomes.