Problem 17 - Discrete Mathematics with Applications

Question

Evaluate each boolean expression, where \(a=2, b=3, c=5,\) and \(d=7\). $$\sim[(a>b) \vee(b \leq d)]$$

Step-by-Step Solution

Answer

The boolean expression \(\sim[(2>3) \vee(3 \leq 7)]\) evaluates to False.

Step 1: Replace variables with values

Let's first replace the given variables \(a\), \(b\), \(c\), and \(d\) with their corresponding values: \(\sim[(2>3) \vee(3 \leq 7)]\)

Step 2: Evaluate comparison operators

Now, let's evaluate the comparison operators: - \(2 > 3\) is False since 2 is not greater than 3 - \(3 \leq 7\) is True since 3 is less than or equal to 7 So, the expression becomes: \(\sim[\text{False} \vee \text{True}]\)

Step 3: Apply "or" operator

Let's apply the "or" operator. An "or" operator returns True if at least one of the operands is True. \(\text{False} \vee \text{True}\) is True So, the expression becomes: \(\sim \text{True}\)

Step 4: Negate the result

Finally, let's negate the result. The negation of True is False: \(\sim \text{True}\) is False Thus, the boolean expression evaluates to False.

Key Concepts

Discrete Mathematics and Boolean Algebra

Discrete mathematics plays a vital role in computer science and engineering, particularly in the arena of algorithms and data structures. An indispensable part of this discipline is boolean algebra. Boolean algebra is the study of truth values (true and false) and how they relate through logical operators. It's instrumental in designing digital circuits and computer algorithms. For instance, when a computer needs to make a decision based on certain conditions, boolean expressions are used to evaluate these conditions.

Understanding how to manipulate and evaluate boolean expressions is foundational. It involves assessing expressions composed of boolean values and operators, often using values assigned to variables. As in the exercise provided, the true essence of evaluation is to simplify the expression systematically, step by step, until we arrive at a conclusive truth value.

Logical Operators

Logical operators are the building blocks of boolean expressions. They include 'and' (conjunction, denoted as \(\wedge\)), 'or' (disjunction, denoted as \(\vee\)), and 'not' (negation, denoted as \(\sim\)). Each operator follows specific rules:

• The 'and' operator returns true if both operands are true.
• The 'or' operator returns true if at least one operand is true.
• The 'not' operator simply inverts the truth value.

In the given exercise, we encounter the 'not' and 'or' operators as part of the boolean expression. Their evaluation is carried out by first understanding the truth value of the operands and then applying the operator rules to those values. A common mistake is to overlook the precedence order; 'not' has the highest precedence, followed by 'and', and then 'or'. Therefore, understanding how these operators work and interact is essential in evaluating expressions accurately.

Comparison Operators

Comparison operators are another category of operators crucial to forming boolean expressions. These include 'greater than' (denoted as >), 'less than' (denoted as <), 'equal to' (denoted as ==), ‘not equal to’ (denoted as !=), ‘greater than or equal to’ (denoted as \(\geq\)) and ‘less than or equal to’ (denoted as \(\leq\)). They are used to compare numerical or sometimes other types of values and produce a boolean result, true or false. This is demonstrated in the provided exercise where \(2>3\) evaluates to false and \(3\leq7\) evaluates to true.

It's vital for students to remember that comparison operators help determine the flow of logic in a program or expression. Misunderstanding these operators can lead to incorrect boolean evaluation. Therefore, students should be comfortable with their semantic meaning and their use in different contexts to strengthen their approach to solving boolean expressions.