Problem 15 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The short answer to the given problem is: Applying De Morgan's law and substituting the given values, we have: \((\sim (5 < 2)) \vee (\sim (3 < 2)) = (True) \vee (True) = True\) Thus, the simplified expression evaluates to \(True\).

Step 1: Write down the given boolean expression

We are given the following boolean expression: $$ \sim|(y < x) \wedge(z < x)| $$

Step 2: Apply De Morgan's law to simplify the expression

We will apply De Morgan's law for AND operations, which states that $$ \sim(A \wedge B) \equiv (\sim A) \vee (\sim B) $$ Applying this law to our given expression, we get: $$ \sim| (y < x) \wedge (z < x) | \equiv (\sim (y < x)) \vee (\sim (z < x)) $$

Step 3: Substitute the given values

Now, let's substitute the given values of \(x=2\), \(y=5\), and \(z=3\) to the simplified expression: $$ (\sim (5 < 2)) \vee (\sim (3 < 2)) $$

Step 4: Evaluate the comparison operations

Next, we will evaluate the comparison operations inside the parentheses: $$ (\sim (False)) \vee (\sim (False)) $$

Step 5: Evaluate the NOT operations

Now, we will evaluate the NOT (\(\sim\)) operations: $$ (True) \vee (True) $$

Step 6: Evaluate the OR operation

Finally, we will evaluate the OR (\(\vee\)) operation: $$ True $$ The result of the boolean expression is \(True\).

Key Concepts

Boolean expressions

Boolean expressions are formulas used in mathematics and computer science to represent logical statements. They consist of variables and logical operators. A Boolean expression evaluates to a Boolean value, namely either true or false. In this context, Boolean expressions are analogous to yes or no questions.

• Variables: These can represent values that are compared or computed. In our exercise, we have variables like \(x\), \(y\), and \(z\).
• Logical operators: These are symbols or words used to connect or modify Boolean expressions, such as AND (\(\wedge\)), OR (\(\vee\)), and NOT (\(\sim\)).
• Evaluation: The process of determining a Boolean expression's truth value based on the values or variables within it.

In our given exercise, we started with a Boolean expression and evaluated it step-by-step using these logical tools and operators, in conjunction with a set of given values.

Logical operators

Logical operators are the building blocks of Boolean expressions. They help define how values in expressions are combined to produce a singular true or false outcome. In our problem, we dealt with three primary logical operators:

• AND (\(\wedge\)): This operator returns true if and only if both operands are true. It acts like a logical glue binding statements together.
• OR (\(\vee\)): This operator is true if at least one operand is true. It provides a way to say "either/or" in logical terms.
• NOT (\(\sim\)): This operator inverts the truth value of its operand. If a statement was true, NOT makes it false and vice-versa.

Understanding these operators is crucial for parsing or simplifying Boolean expressions, as seen when De Morgan's laws were applied to simplify \(\sim(|(y < x) \wedge (z < x)|)\) using these three operators.

Discrete mathematics

Discrete mathematics is a branch of mathematics dealing with distinct and separate objects. It's extensively used in computer science to solve problems involving networks, algorithms, cryptography, and of course, Boolean logic. The prior exercise forms the backbone of many applications in discrete tasks:

• Logical reasoning: A core aspect of discrete mathematics; it teaches structured thought processes using logical operators.
• Integer relationships: Problems often deal with definite, countable elements, such as the integer variables we saw in the expression.
• Application of De Morgan's laws: These laws transformed our complex AND operator situations into simpler OR scenarios, making computations manageable.

Discrete mathematics emphasizes a "binary" way of problem-solving, akin to true/false scenarios prevalent in Boolean expressions, and it forms a critical basis for logical problem solving in computing scenarios.