Problem 51 - Discrete Mathematics with Applications

Question

Evaluate each boolean expression. $$(0 \uparrow 1) \downarrow(0 \uparrow 1)$$

Step-by-Step Solution

Answer

Answer: 1

Step1: Evaluate OR expressions in parentheses

To evaluate the OR expressions in the parentheses, we need to recall the OR operation's truth table which states that OR expression will be true (1) if at least one of the operands is true (1) otherwise it will be false (0). For our expression, \((0 \uparrow 1)\), we identify that one of the operands (1) is true. Thus, the OR expression evaluates to 1. The resulting expression after evaluating the parentheses will be: $$ (1) \downarrow (1) $$

Step 2: Evaluate the AND expression

Now, let's evaluate the remaining AND expression. AND operation is denoted as '\(\downarrow\)'. We need to recall the AND operation's truth table, which states that AND expression will be true (1) if both operands are true(1), otherwise, it will be false (0). For our expression, \((1 \downarrow 1)\), we observe that both operands are true (1). Thus, the AND expression evaluates to 1. Therefore, the final result of the given boolean expression is: $$ 1 $$

Key Concepts

Boolean Expressions

Boolean expressions are logical statements that can either be true or false. These expressions use a variety of operators, such as NOT, AND, OR, NAND, NOR, XOR, and XNOR. Each of these operators helps in forming complex logic statements that computer systems use to make decisions and perform calculations.

For instance, in the problem given, the expression \[(0 \uparrow 1) \downarrow(0 \uparrow 1)\]is a Boolean expression. Here, we first evaluate portions of this expression to find its overall truth value. Recognizing and manipulating these expressions allows us to simplify logic circuits in digital electronics.

Logical OR Operation

The logical OR operation is a fundamental concept within Boolean algebra. It is represented by the symbol "\( \cup \)". In Boolean expressions, this operation evaluates to true if at least one of its operands is true.

In terms of real-world application, imagine a scenario where you need to determine if someone will pass a course if they achieve a certain grade in either coursework or the final exam. Here, OR logic dictates that as long as one of the criteria is met, the output will be true (pass).The truth table for the logical OR illustrates this concept well:

• \( 0 \cup 0 = 0 \)
• \( 0 \cup 1 = 1 \)
• \( 1 \cup 0 = 1 \)
• \( 1 \cup 1 = 1 \)

In the exercise, the operation \[(0 \cup 1)\]evaluates to 1 because one operand is true. This example highlights how OR can simplify decision-making processes.

Logical AND Operation

The logical AND operation is another key component of Boolean algebra, typically represented by the symbol "\( \cap \)". This operation is only true if both operands involved in the operation are true. If either operand is false, the entire AND operation evaluates to false.

Consider using an AND operation in a real-life scenario such as deciding to wear a jacket only if it's both cold and raining. Here, both conditions must be true for the outcome (wearing the jacket) to occur.To better understand this, let's look at the truth table for the logical AND operation:

• \( 0 \cap 0 = 0 \)
• \( 0 \cap 1 = 0 \)
• \( 1 \cap 0 = 0 \)
• \( 1 \cap 1 = 1 \)

In the expression \[(1 \cap 1)\], both operands are true, resulting in an output of 1. This shows how specific requirements must be fulfilled for an AND operation to result in true.

Truth Tables

Truth tables are a helpful tool for visualizing how Boolean operations work. They clearly show all possible combinations of inputs and their corresponding outputs, making it easier to understand how logical expressions function.

A truth table for a single operation lists all input combinations of 0s and 1s (representing false and true), and the resulting output for each combination. It is essential in analyzing complex logical expressions, especially when working with digital circuits and computer programs. For example, with the OR operation discussed earlier, a truth table shows you exactly when the result will be true versus false based on the inputs. This simplification aids designers in predicting and verifying circuit behavior efficiently. Similarly, AND operation truth tables help easily visualize which scenarios will satisfy all conditions for an output of true. These representations form the backbone of logical reasoning in digital computing.