Problem 4 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

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

Step 1: Apply the Distributive Law

We will apply the distributive law and group the terms accordingly. \(xy + x'y' + xy' = (xy + xy') + x'y'\)

Step 2: Apply the Absorption Law

Next, we will apply the absorption law. According to the absorption law, \(x + xy = x\), so the expression can be further simplified. \((xy + xy') + x'y' = x(y + y') + x'y'\)

Step 3: Use the complementary law

According to the complementary law, \(y + y' = 1\). We will replace \(y + y'\) with 1. \(x(y + y') + x'y' = x(1) + x'y'\)

Step 4: Simplify the expression

Now, simplify the expression by removing any unnecessary terms. \(x(1) + x'y' = x + x'y'\) The simplified boolean expression is \(x + x'y'\).

Key Concepts

Distributive Law in Boolean Algebra

In Boolean Algebra, the Distributive Law allows us to rearrange and factor expressions in a way that might be familiar from regular algebra, but with a twist. The law can be expressed as:

• For operations involving AND over OR: \(a(b + c) = ab + ac\)
• For operations involving OR over AND: \(a + (bc) = (a + b)(a + c)\)

By applying the Distributive Law, we can simplify complicated Boolean expressions by breaking them into smaller parts or grouping terms for easier manipulation. In our example,
we transformed the expression \(xy + xy' + x'y'\) into \((xy + xy') + x'y'\).
This grouping prepares the expression for further simplification using other laws, such as the Absorption Law.

Absorption Law

The Absorption Law is a powerful tool in Boolean Algebra that simplifies expressions by "absorbing" terms. The essence of this law is seen in the equations:

• \(a + ab = a\) - the OR Absorption Law
• \(a(a + b) = a\) - the AND Absorption Law

The idea is that a term can "absorb" another if it fully includes the other term's factors, hence reducing redundancy.
In the given solution, we applied the Absorption Law to \((xy + xy') + x'y'\) by recognizing that \((xy + xy')\) simplifies using \(x(y + y')\),
and because \(x + xy\) can be written directly as \(x\), where commonalities between terms are "absorbed."
Thus, the expression continues to be reduced, allowing for simpler evaluation and further manipulation.

Complementary Law

The Complementary Law addresses how variables and their complements relate to each other in Boolean Algebra. This law states:

• \(a + a' = 1\)
• \(a \cdot a' = 0\)

These identities arise from the logic that a statement and its opposite cover all possible outcomes, hence always results in a true condition,
which is why \(a + a' = 1\). Conversely, a statement AND its complement is always false, as a condition can't be true and false simultaneously, resulting in zero.
In our solution, the Complementary Law was crucial for simplifying \(x(y + y') + x'y'\) because \(y + y' = 1\). By replacing \(y + y'\) with 1,
the expression \(x(y + y')\) simply becomes \(x(1)\), or just \(x\).
These laws are foundational tools that help in systematically pulling apart and reducing complex Boolean expressions easily.