Truth Tables

Truth Tables Basics

  • Truth tables are a mathematical table used to determine if a logical expression is true or false.
  • They are used to represent the output results of all possible inputs in logical operations.
  • They are particularly useful when dealing with multiple conditions or complex logical statements.

Components of a Truth Table

  • A truth table has a row for every possible combination of inputs.
  • Each row contains one possible scenario of inputs and the resulting output for that scenario.
  • The inputs are listed in the first few columns, followed by a column for each logical operator used in the expression.
  • The last column typically shows the result of the whole logical expression.

Reading Truth Tables

  • To read a truth table, start from the left and consider each row’s input values. Interpret True as 1 and False as 0.
  • Applying the specified logical operations to these inputs should result in the output shown to the right.

Building Truth Tables

  • To build a truth table, begin by deciding which logical operators to feature (for example, AND, OR and NOT) and then specify all the possible combinations of inputs.
  • Next, fill in the table by calculating the output for each row.
  • Be aware that for two inputs, you will have four possible scenarios, and hence four rows (00, 01, 10, 11). However, for three inputs, combinations increase to eight, hence eight rows.

Effects of Logical Operations on Truth Tables

  • The AND operation only results in a True value when both inputs A and B are True.
  • The OR operation results in a True value if either A, B or both are True.
  • The NOT operation simply inverts the current state from True to False or vice versa.

Troubleshooting with Truth Tables

  • If your logic expression doesn’t appear to give correct results, use a truth table to verify each stage of the processing.
  • Remember, order in logical operations is important. Ensure you follow the precedence rules.

To put this all into practice, try creating your own truth tables using various combinations of logical operators, and validate them by checking your results across the row with the logical formula.