Problem 14 - Discrete Mathematics with Applications

Question

Use De Morgan's laws to evaluate each boolean expression, where \(x=2\) \(y=5,\) and \(z=3\) $$\sim |(x

Step-by-Step Solution

Answer

The given Boolean expression, after substituting the values of x, y, and z and applying De Morgan's law, simplifies to: \(false \vee true\). As one of the operands is true, the expression evaluates to \(true\).

Step 1: Write down the given Boolean expression

The given Boolean expression is: \(\sim |(x < z) \wedge (y < z)|\)

Step 2: Apply De Morgan's law

Using De Morgan's first law, we can rewrite the given expression as: \(\sim(x < z) \vee \sim(y < z)\)

Step 3: Substitute values for x, y, and z

Now we substitute the given values for x, y, and z: \(\sim(2 < 3) \vee \sim(5 < 3)\)

Step 4: Evaluate the inequality in each term

Let's evaluate each inequality: \(\sim(true) \vee \sim(false)\)

Step 5: Evaluate the negations

Now we find the negation of each term: \(false \vee true\)

Step 6: Evaluate the Boolean expression

Finally, evaluate the Boolean expression using the OR (\(\vee\)) operation: Since one of the operands is true, the whole expression evaluates to true. Thus, the value of the given Boolean expression is \(true\).

Key Concepts

Boolean Algebra

Boolean algebra is a branch of algebra that deals with variables that have two possible values: true and false, often denoted as 1 and 0, respectively. It's a mathematical way to represent the logic of yes-no questions, making it essential in the design and operation of computer circuits and other digital systems.

At the core of Boolean algebra are logical operations that express how multiple boolean values are combined. The primary operations are AND (conjunction), which is only true if both operands are true; OR (disjunction), true if at least one operand is true; and NOT (negation), which reverses the truth value of its operand. These operations follow specific laws and rules that allow complex logical expressions to be simplified or reformulated for clarity, which is particularly useful in computer science and electronic engineering.

Negation in Boolean Logic

Negation in Boolean logic is the operation that inverts the value of a boolean expression. If we consider a boolean variable 'A', which can be either true or false, the negation of 'A' (denoted as \( \sim A \)) will be false if 'A' is true, and true if 'A' is false. This is analogous to saying 'not A' in everyday language.

This operation is crucial for constructing logical statements and is foundational to De Morgan's laws. These laws assert that the negation of a conjunction (\( \sim(A \wedge B) \)) is equivalent to the disjunction of the negations (\( \sim A \vee \sim B \) ) and vice versa. Applying negation properly can lead to a better understanding of complex logical structures and provide a means to simplify expressions in Boolean algebra.

Logical Disjunction

Logical disjunction, represented by the OR operator (\( \vee \)), is a key operation in Boolean algebra where the result is true if at least one of the operands is true. If we have two boolean expressions, 'A' and 'B', the disjunction ('A OR B') will be true if 'A' is true, or 'B' is true, or if both 'A' and 'B' are true.

In the context of logical problems and computer science, disjunction allows us to build conditions that are satisfied under multiple circumstances. For instance, if a security system triggers an alarm when either a door opens (\( A \)) or a window breaks (\( B \)), the alarm state can be represented as \( A \vee B \). Disjunction is a foundational block along with conjunction and negation in Boolean calculations, allowing for the modeling of various logical scenarios in digital systems.