Contents
In this publication, we will consider 5 logical operations: we will give their definitions, recording methods (if any), as well as their corresponding truth tables.
Basic terms
P’C ‹СЃРєР ° Р · С‹ РІР ° РЅРёРµ – a sentence that expresses a certain judgment, by which it is determined whether it is true (indicated by the number “1”) or false (written as “0”).
Logical operation – an action in the mind, as a result of which the content or scope of concepts changes, as well as new concepts are formed.
boolean expression – an oral or written statement in which there are both constants and variables. Depending on the values that the variables take, the boolean expression can be either true (1) or false (0).
Complex boolean expression is a compound expression that includes one or more simple and/or complex logical expressions linked by logical operations.
Logical operations and truth tables
Multiplication (conjunction)
Conjunction is a complex logical expression that is true exceptional if both simple expressions of which it consists are true. Otherwise, it is false.
It is designated as follows: F = A & B.
Truth table for conjunction
| A | B | F |
| 1 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 0 |
Addition (disjunction)
Disjunction is a complex logical expression that is considered true if at least one of the simple logical expressions is true. Accordingly, it is false only if both simple expressions are also false.
It is written like this: F = A + B.
Truth table for disjunction
| A | B | F |
| 1 | 1 | 1 |
| 1 | 0 | 1 |
| 0 | 1 | 1 |
| 0 | 0 | 0 |
Negation (inversion)
Inversion is a complex logical expression obtained as follows:
- If the original logical expression is true, then the result of the negation is considered false.
- If the original expression is false, then the result of negation is true.
Truth table for inversion
| A | notA |
| 1 | 0 |
| 0 | 1 |
Following (implication)
Implication is a complex logical expression that connects two simple expressions. In this case, the first is a condition (A), and the second is a consequence of (B). The expression is always true, except when true implies false.
Truth table for implication
| A | B | F |
| 1 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 1 |
| 0 | 0 | 1 |
Equivalence (equivalence)
Equivalence – a complex logical expression that is considered true only if both simple expressions are equally true.
Truth table for equivalence
| A | B | F |
| 1 | 1 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
Note: The operations in a complex boolean expression are performed in the following order:
- Inversion
- Conjunction
- Disjunction
- Implication
- Equivalence
Parentheses must be used to reverse this order.