Problem 1 - Discrete Mathematics with Applications

Question

Simplify each boolean expression using the laws of boolean algebra. $$x y+x y^{\prime}$$

Step-by-Step Solution

Answer

The given boolean expression is \(x y + x y'\). Applying the distributive law in reverse, we factor the expression: \(x(y + y')\). Since \(y+y' = 1\), we substitute the result and get \(x (y+y') = x (1)\). Finally, using identity law, we simplify the expression to \(x(1) = x\). Thus, the simplified boolean expression is \(x\).

Step 1: Apply the Distributive Law

We will first apply the distributive law to the given expression. The distributive law states that for any variables A, B, and C, we have \(A (B + C) = AB + AC\). By applying this law in reverse, we can factor the expression as follows: \(x y + x y' = x(y + y')\)

Step 2: Evaluate y + y'

Given that \(y\) and \(y'\) are complements of each other, they follow the inverse law. The inverse law states that for any variable A, we have \(A + A' = 1\). In our case, that would be: \(y + y' = 1\)

Step 3: Substitute the result of Step 2

Now, we will substitute the result from Step 2 into the expression from Step 1: \(x (y + y') = x (1)\)

Step 4: Apply the Identity Law

For any variable A, the identity law states \(A \cdot 1 = A\). Applying this law to our expression, we have: \(x (1) = x\)

Step 5: Conclusion

The simplified boolean expression for \(x y + x y'\) is \(x\).

Key Concepts

Distributive Law

In Boolean algebra, the Distributive Law is similar to its arithmetic counterpart but tailored for logical operations. It helps in simplifying expressions by factoring common terms. Imagine you have an expression like this:

• \( A(B + C) = AB + AC \)

When you see common terms, you can factor them out, much like factoring an equation in traditional algebra.
In the original problem, we started with \( x y + x y' \). Recognizing \( x \) as a common factor, we applied the Distributive Law to "factor it out," transforming the expression into \( x(y + y') \). This is done because it allows us to simplify expressions by dealing with fewer operations.
This law is fundamental as it aids in reducing the complexity of logical expressions, making them easier to manage and understand.

Inverse Law

The Inverse Law in Boolean algebra relates to how a variable interacts with its complement. Think of complement as the opposite, like on and off states. This law states that for any Boolean variable \( A \), the sum with its complement equals one:

• \( A + A' = 1 \)

In the simplified exercise, we dealt with \( y \) and its complement \( y' \). Hence, upon encountering \( y + y' \), we directly apply the Inverse Law, concluding that \( y + y' = 1 \). This step dramatically simplifies the expression since any variable added to its complement yields a consistent result of one.
Recognizing complements is crucial for effective simplification of Boolean expressions since it allows idle parts of expressions to be replaced with constants that significantly reduce their complexity.

Identity Law

The Identity Law in Boolean algebra states that multiplying a variable by one doesn't change its value. This rule mirrors the arithmetic identity for multiplication where any number multiplied by one remains the same. In Boolean terms, for a variable \( A \), the Identity Law is expressed as:

• \( A \cdot 1 = A \)

In our given problem, after applying the Distributive and Inverse Laws, we were left with \( x(1) \). According to the Identity Law, \( x \times 1 \) is simply \( x \). This principle assures that multiplying by the Boolean "1" is effectively neutral in terms of value, allowing for another layer of simplification.
By understanding the Identity Law, one can efficiently trim Boolean expressions by removing redundant or non-impactful elements, ensuring the solution is as compact and clear as possible.