Problem 16 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The short answer to the given boolean expression is: \(\ [\sim(b< c)] \wedge[\sim(c< d)] = False \).

Step 1: Replace Variables

\(a = 2, b = 3, c = 5, d = 7\) So, the expression becomes: \( [\sim(3< 5)] \wedge [\sim(5< 7)] \)

Step 2: Evaluate Boolean Expressions

Now, let's evaluate the boolean expressions inside the square brackets. \( (3 < 5) = True \) and \( (5 < 7) = True \) Now we now the expression: \[ [\sim(True)] \wedge [\sim(True)] \]

Step 3: Apply the NOT Operator

Now, we can apply the NOT operator (∼) to the boolean expressions. \( \sim(True) = False \) So, the expression becomes: \[ [False] \wedge [False] \]

Step 4: Evaluate Final Expression using the AND Operator

Finally, let's evaluate the expression using the AND operator (∧). \[ False \wedge False = False \] So, the evaluation of the given boolean expression is: \[ [\sim(b< c)] \wedge[\sim(c< d)] = False \]

Key Concepts

Boolean Expressions

Boolean expressions are a fundamental part of Boolean algebra, used extensively in computer science and digital logic. These expressions are statements that result in a Boolean value, which means they either evaluate to "True" or "False." In the context of the exercise above, Boolean expressions are evaluated step-by-step using given variable values.

• First, the given mathematical expressions inside the brackets, such as \((b < c)\) and \((c < d)\), are computed. Here, they compare numbers with relational operators, yielding Boolean results. \((3 < 5)\) results in "True," while \((5 < 7)\) also returns "True."
• Next, logical negations and combinations use Boolean operators, which shift these initial results from "True" to "False" or combine them to evaluate their truthiness altogether.

Understanding these expressions is crucial as they form the basis for decision-making processes in programs and electronic circuits.

Logical Operators

Logical operators process Boolean expressions by performing calculations, much like arithmetic operators do with numbers. They help in forming complex Boolean expressions that produce truth values. Some of the common logical operators include:

• **NOT (∼):** This operator reverses the truth value of a Boolean expression. For instance, when applied to "True," the NOT operator outputs "False."
• **AND (∧):** This operator returns "True" if both operands are "True." With the expression \([False] \wedge [False]\), as in the exercise above, the result is "False" because both operands are false.
• **OR:** While not shown in the exercise, it's worth noting that the "OR" operator returns "True" if at least one operand is "True."

Logical operators are integral in constructing complex conditions and contributing to the functioning of control flows, which direct the execution path within programs.

Truth Values

Truth values are the possible outcomes of evaluating Boolean expressions. In Boolean algebra, there are only two truth values: "True" and "False." They are akin to binary numbers, simplifying many digital systems that operate with 0s and 1s.

• Truth values emerge after evaluating relational or logical expressions, as demonstrated in the initial steps of the exercise where \((3 < 5)\) yields "True."
• Further operations, like negations or logical combinations, can alter these values. For example, applying the NOT operator to a "True" results in "False," changing the overall outcome.

Truth values are essential for computers and digital devices, enabling decisions and functions based on logical conditions which ultimately fuel operations, such as running algorithms, processing instructions, and conducting tasks based on conditions.