Problem 42 - Discrete Mathematics with Applications

Question

Determine whether or not the assignment statement \(x \leftarrow x+1\) will be executed in each sequence of statements, where \(i \leftarrow 2, j \leftarrow 3, k \leftarrow 6,\) and \(x \leftarrow 0\). $$ \begin{array}{l} \text { If }(i<3) \wedge(j<4) \text { then } \\ \qquad x \leftarrow x+1 \end{array} $$ else $$ y

Step-by-Step Solution

Answer

The given condition is \((i<3) \wedge (j<4)\) with \(i=2\) and \(j=3\). Evaluating the condition, we have \((2<3) \wedge (3<4)\), which simplifies to \(True \wedge True\), and this is True. Since the condition is true, the assignment statement \(x \leftarrow x + 1\) will be executed in this sequence of statements.

Step 1: Identify Variables Values

The problem provides initial values for each variable: \(i = 2, j = 3, k = 6\), and \(x = 0\). Step 2: Examine the condition in the 'if' statement

Step 2: Examine Condition

The given condition to evaluate is: \((i<3) \wedge (j<4)\). According to the variable values, we have \((2<3) \wedge (3<4)\). Step 3: Evaluate the condition

Step 3: Evaluate Condition

Evaluate both parts of the condition separately: \(2<3\) is true, and \(3<4\) is also true. Step 4: Determine the outcome of the 'if' statement

Step 4: If Statement Outcome

Considering the results of the previous evaluations, we can conclude the AND operation: \(True \wedge True\) is True. Since the condition evaluates to true, we execute the first statement in the 'if' block: \(x \leftarrow x + 1\) Therefore, the assignment statement \(x \leftarrow x + 1\) will be executed in this sequence of statements.

Key Concepts

Conditional Statements

In programming and mathematics, conditional statements play a crucial role in decision-making. They allow a program to choose different paths based on certain conditions, much like making decisions in everyday life.
In the given exercise, the conditional statement is formulated using an **if-else** structure. This structure checks whether a specific condition is true or false.
Here’s how it works:

• If the condition is true, the program executes the code block within the 'if'.
• If the condition is false, the program executes the alternative code block in the 'else'.

In our exercise, the condition checks whether both

• \(i < 3\)
• \(j < 4\)

are true. Since this composite condition is true (as both sub-conditions are true), the statement in the 'if' block is executed.
Understanding conditional statements helps in constructing logical sequences and prompts better problem-solving strategies.

Boolean Logic

Boolean logic is a form of algebra in which the values of the variables are true or false. It is fundamental to computer science, digital electronics, and mathematics. In our exercise, the 'AND' operator, symbolized by \(\wedge\), is a type of Boolean operator that requires both operands to be true for the entire expression to be true.
Here's how the 'AND' operator works in Boolean logic:

• **True \(\wedge\) True = True**
• **True \(\wedge\) False = False**
• **False \(\wedge\) True = False**
• **False \(\wedge\) False = False**

For the exercise, evaluating \((i<3) \wedge (j<4)\) results in **True \(\wedge\) True**, hence the overall condition is true.
This outcome dictates whether the statements inside the corresponding 'if' or 'else' block are executed. Mastery of Boolean logic is essential for designing algorithms and understanding code execution.

Mathematical Reasoning

Mathematical reasoning is the process of drawing logical conclusions based on premises or known facts. It involves making deductions to uncover unknown information. In our exercise, the reasoning process involves interpreting the conditions and inferencing correctly.
Here's how reasoning facilitated the solution:

• Given initial values, substitute them into the condition \((i<3) \wedge (j<4)\).
• Analyze each part separately:
  • \(i < 3\) becomes \(2 < 3\), which is true.
  • \(j < 4\) becomes \(3 < 4\), true again.

Combining these evaluations using Boolean logic results in an outcome that informs execution of the relevant code path. Mathematical reasoning thus helps solve problems efficiently, ensuring accuracy and logical flow in mathematics and computational tasks.