Equality Operation

The equivalence operation (equality or biconditional) returns a true value (1) if and only if both of its operands have the same value. It is represented by the symbol ↔. $$ A \leftrightarrow B $$

This is one of the fundamental operations in Boolean algebra and propositional logic.

The equivalence operation between two Boolean variables \( A \) and \( B \) is defined as follows:

$$ A \leftrightarrow B $$

This expression is true (1) if \( A \) and \( B \) are both true or both false. In other words, \( A \leftrightarrow B \) is true if \( A \) and \( B \) are logically equivalent.

The truth table for the equivalence operation is as follows:

\[
\begin{array}{|c|c|c|}
\hline
A & B & A \leftrightarrow B \\
\hline
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0 \\
1 & 1 & 1 \\
\hline
\end{array}
\]

To better understand the equivalence operation, let's consider a practical example:

 If \( A \) represents "It is daytime" and \( B \) represents "The sun is visible", \( A \leftrightarrow B \) would be true if both statements are true (it is daytime and the sun is visible) or both are false (it is not daytime and the sun is not visible).

The Relationship Between Equivalence and Boolean Logical Operators

If one statement is true and the other is false, the operation \( A \leftrightarrow B \) would be false.

The equivalence operation can be expressed using other fundamental logical operations such as AND, OR, and NOT. One equivalent form is:

\[ A \leftrightarrow B = (A \land B) \lor (\neg A \land \neg B) \]

This expression indicates that \( A \leftrightarrow B \) is true if both variables are true or both are false.

The equivalence operation is applied in various fields. For instance, in digital circuits, it is used to verify if two logical signals are identical. In propositional logic, it helps to formulate and solve complex logical expressions. In algorithms, it is used to perform logical comparisons between variables.

Properties of Equivalence

The equivalence operation satisfies the following properties:

• Commutativity
  \[ A \leftrightarrow B = B \leftrightarrow A \]
• Associativity
  \[ (A \leftrightarrow B) \leftrightarrow C = A \leftrightarrow (B \leftrightarrow C) \]
• Idempotence
  \[ A \leftrightarrow A = 1 \]
• Involution
  \[ \neg (A \leftrightarrow B) = A \leftrightarrow \neg B \]