Problem 44 - 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 \(x \leftarrow 0\). While \(\sim(i \leq j)\) do begin \(x \leftarrow x+1\) \(i \leftarrow i+1\) endwhile

Step-by-Step Solution

Answer

In conclusion, the assignment statement \(x \leftarrow x+1\) will not be executed in the given sequence of statements, as the loop condition \(\sim(i \leq j)\) is false from the beginning.

Step 1: Identify the initial values of each variable

Begin by identifying the initial values of each variable. They are given as follows: \(i \leftarrow 2, j\leftarrow 3, k\leftarrow 6,\) and \(x\leftarrow 0\). These initial values mean that at the beginning: \( i = 2, j = 3, k = 6 \text{ and } x = 0 \)

Step 2: Analyze the while loop condition

The given condition is \(\sim(i \leq j)\). This means that the while loop will continue executing while this condition evaluates to true. The loop will stop the moment this condition evaluates to false (i.e., when \(i \leq j\)).

Step 3: Analyze the increment of variables

Inside the while loop, we have two assignment statements: \( x \leftarrow x+1 \) and \( i \leftarrow i+1 \). These statements mean that: - x is incremented by 1 for each iteration in the while loop. - i is also incremented by 1 for each iteration in the while loop.

Step 4: Analyzing the loop's execution

In the beginning, \(i=2\) and \(j=3\). Since \(i \leq j\), the negation of this condition, \(\sim(i\leq j)\), is false. Since the loop condition is false, the while loop will not execute, and the assignment statement \(x \leftarrow x+1\) will not be executed. In conclusion, the assignment statement \(x \leftarrow x+1\) will not be executed in the given sequence of statements.

Key Concepts

Conditional Statements

Conditional statements are used in programming to control the flow of a program based on certain conditions. They are essential for decision-making and allow a program to execute different code paths based on the evaluation of a condition. In the example exercise, the condition in the while loop is \(\sim(i \leq j)\). This is a negation of the condition \(i \leq j\), meaning it checks if \(i > j\).
Understanding how a conditional statement works involves knowing:

• Logical operators: These include \(\leq\), \(=\), \(>\), etc., which help test expressions.
• Negation: Represented by the symbol \(\sim\), this changes true to false and vice versa.

In this scenario, since \(i = 2\) and \(j = 3\), the statement \(i \leq j\) is true, meaning \(\sim(i \leq j)\) becomes false. Thus, the loop will not execute, as the conditional statement is false from the start.

Loop Conditions

Loop conditions determine how many times a loop will iterate. They act as gatekeepers for looping constructs like "while" loops. For a loop to keep running, the condition must evaluate to true.
In a while loop, the condition is evaluated before each iteration of the loop. A while loop continues to execute its block of code as long as the specified condition is true. The condition \(\sim(i \leq j)\) is evaluated at the start of each iteration in our example.

• If a condition is false, the code inside the loop will never run.
• If initially false, like our case, the loop is skipped altogether.
• The condition is crucial for determining the loop's number of executions.

In the context of the given exercise, the loop's condition is false from the beginning; hence, the loop body - where the increment happens - is never reached.

Variable Initialization

Variable initialization is the process of assigning a beginning value to a variable before it is used in a program. Initial values are necessary for many computations to avoid undefined behavior or errors.
In our exercise, initial values ensure that the logic in the loop starts with known states. Here, the variables are initialized as follows:

• \(i = 2\)
• \(j = 3\)
• \(k = 6\) (Though unused in the loop's logic)
• \(x = 0\)

With initialization:

• We ensure that the loop conditions have proper values to evaluate.
• Knowing initial values allows us to accurately predict the execution flow, as seen when calculating whether \(x\) will increment.

Since the loop condition is false, the initial value of \(x\) remains unchanged throughout the program, emphasizing how essential initialization is for following control logic.