Question

Show that \(\mathrm{x} \wedge \mathrm{y}\) is logically equivalent to \(\mathrm{y} \wedge \mathrm{x}\).$$ \begin{array}{|c|c|c|c|} \hline(1) & (2) & (3) & (4) \\ \hline \mathrm{x} & \mathrm{y} & \mathrm{x} \wedge \mathrm{y} & \mathrm{y} \wedge \mathrm{x} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \end{array} $$

Short answer

The truth tables for x ∧ y and y ∧ x are as follows: $$ \begin{array}{|c|c|c|c|} \hline (1) & (2) & (3) & (4) \\ \hline \mathrm{x} & \mathrm{y} & \mathrm{x} \wedge \mathrm{y} & \mathrm{y} \wedge \mathrm{x} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \end{array} $$ As the outputs of x ∧ y and y ∧ x are the same for all input values, they are logically equivalent.

Step-by-step solution

Define the logical AND operation

When working with Boolean Algebra, there are a few basic operations - one of which is the AND operation, denoted as ∧. The AND operation takes two input values (x and y) and returns True only if both input values are True. Otherwise, it returns False. The truth table for the AND operation is: $$ \begin{array}{|c|c|c|} \hline \mathrm{x} & \mathrm{y} & \mathrm{x} \wedge \mathrm{y} \\\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} \\\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} \\\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\\ \hline \end{array} $$

Construct a truth table for x ∧ y and y ∧ x

Now that we know the definition of the AND operation, we can create a truth table to compare the expressions x ∧ y and y ∧ x. We already have the table for x ∧ y, so we only need to create the table for y ∧ x: $$ \begin{array}{|c|c|c|} \hline \mathrm{y} & \mathrm{x} & \mathrm{y} \wedge \mathrm{x} \\\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} \\\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} \\\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\\ \hline \end{array} $$

Compare the tables and determine the equivalence

Let's compare the columns corresponding to x ∧ y and y ∧ x: $$ \begin{array}{|c|c|c|c|} \hline (1) & (2) & (3) & (4) \\\ \hline \mathrm{x} & \mathrm{y} & \mathrm{x} \wedge \mathrm{y} & \mathrm{y} \wedge \mathrm{x} \\\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\\ \hline \end{array} $$ We can observe that the outputs of the two expressions are the same in each row. This means that the two expressions x ∧ y and y ∧ x are indeed logically equivalent since they produce the same result for all possible input values.

Key concepts

Boolean Algebra

Boolean algebra is a branch of mathematics that deals with values that are either true or false, commonly represented as 1 or 0. It involves operations on these binary values that follow specific rules. One of the foundational principles of Boolean algebra is that it can simplify the way we analyze and design electronic circuits and computer algorithms by using binary variables.

The operations in Boolean algebra have direct analogues in classical algebra, but they operate under different rules. For example, in Boolean algebra, 1 + 1 is not 2; it remains 1 because we interpret it as 'True or True' which is still 'True'. Understanding these operations is crucial for areas like computer science and electrical engineering, where logic circuits and decision-making processes rely on these principles.

AND Operation

The AND operation is a fundamental logical function in Boolean algebra. It is denoted by the symbol \(\wedge\) and is akin to multiplication in classical algebra. The AND operation takes two Boolean inputs and returns a 'True' result only if both inputs are 'True'. If either of the inputs is 'False', the result will be 'False'.

In simplest terms, the AND operation can be compared to a series of switches that must all be on for a circuit to be active. This operation is critical in constructing logical conditions in computer programming where multiple criteria must be met and in designing circuits where various conditions must be satisfied to trigger an action.

Truth Table

A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments. Essentially, it lists all possible input combinations to a logical function and the resulting outputs.

Truth tables are valuable tools for anyone dealing with logical expressions as they provide a clear and concise way to demonstrate how an operation behaves and to verify logical equivalences. By comparing the truth tables of two operations, we can determine if two logical expressions are equivalent, as was demonstrated in the exercise with the AND operation \(x \wedge y\) being shown to be logically equivalent to \(y \wedge x\).