Question

Suppose that \(x, y,\) and \(z\) are int variables, and \(x=10, y=15,\) and \(z=20\) Determine whether the following expressions evaluate to true or false. a. \(\quad !(x>10)\) b. \(x<=5 | 1 \quad y<15\) \(\begin{array}{llllll}\text { c. } & \langle x & !=5) & \& \& \quad(y \quad !=z)\end{array}\) \(\begin{array}{ll}\text { d. } \quad x>=z & || \quad(x+y>=z)\end{array}\) e. \(\quad(x=z) \quad|| \quad(z-2 \quad !=20)\)

Short answer

a. True, b. False, c. True, d. True, e. True.

Step-by-step solution

Analyze Expression a

Expression a is \(!(x > 10)\). Since \(x = 10\), we have \(x > 10\) as false. The NOT operator \(!\) negates the result, so \(!(x > 10)\) is true.

Evaluate Expression b

Expression b is \(x <= 5 || y < 15\). Substitute \(x = 10\) and \(y = 15\). The part \(x <= 5\) is false, and \(y < 15\) is also false. Thus, the entire expression evaluates to false.

Check Expression c

Expression c is \((x != 5) && (y != z)\). Substitute \(x = 10\), \(y = 15\), and \(z = 20\). Both conditions \(x != 5\) and \(y != z\) are true, so the expression evaluates to true.

Examine Expression d

Expression d is \(x >= z || (x + y >= z)\). Substitute \(x = 10\), \(y = 15\), and \(z = 20\). The first part \(x >= z\) is false. The second part \(x + y (which is 25) >= z\) is true, so the expression evaluates to true.

Solve Expression e

Expression e is \((x < y - 2) && (y >= z) || (z - 2 !== 20)\). Substitute \(x = 10\), \(y = 15\), and \(z = 20\). First check \(x < y - 2\) which simplifies to \(10 < 13\), true. Check \(y >= z\) which is false. Now, \(z - 2 = 18\) and \(not equal to 20\), true. Thus, the expression evaluates to true.

Key concepts

C++ expressions

A C++ expression is a series of operands and operators that, when compiled, gives out a value. Expressions can have different components, such as variables, constants, and functions, and use various operators like arithmetic, logical, or bitwise.
In the context of the exercise you provided, each line constitutes a separate C++ expression containing logical operators. These operators evaluate the relationships between integers by comparing their values or states.
For example, in the expression \[!(x > 10)\], the components include a logical NOT operator \(!\) and a relational operator \(>\).

• The expression checks the relationship \(x > 10\) and then negates it using \(!\).
• If the result is false, after negation, it becomes true, highlighting the power of logical operation.

Understanding how to construct and evaluate expressions is crucial in C++, as it forms the basis of programming logic.

Integer variables in C++

Integer variables in C++ are datatypes used to store whole numbers. These can be positive, negative, or zero without decimals. Keywords like `int` are commonly used to declare integer variables.
For example:

• int x = 10;
• int y = 15;
• int z = 20;

The role of integer variables in programming is crucial because they represent discrete data. They're used in loops, counters, logical conditions, and much more.
In the exercise, integer variables \(x, y, ext{ and } z\) are initialized at specific values. These values provide a scenario to evaluate conditions and logical expressions.
Since the values are stored as integers, C++ automatically manages their operations without losing precision, making them ideal for many applications.

Conditional statements in C++

Conditional statements in C++ allow you to make decisions within your code. Using logic, these statements execute certain code blocks based on whether conditions are true or false.
Conditional statements commonly use \( ext{if}, ext{else if}, ext{else}\) structures. This makes your program dynamic and responsive to different input situations.
When dealing with expressions from the exercise, you can imagine using these logical conditions in a broader program context where control flow might depend on the evaluations:

• If the condition \(x <= 5 || y < 15\) is true, run a specific block of code.
• Else, execute an alternative code segment.

Such logic is vital in real-world applications to ensure your program can adapt its behavior based on variable data inputs and specific rules.

Boolean expressions in C++

Boolean expressions evaluate to either true or false, essentially representing a binary choice. In C++, boolean values are represented using the `bool` data type, which can only have `true` or `false`.
Logical operators like `&&` (and), `||` (or), and `!` (not) are primarily used to build complex boolean expressions.
Let’s look at the exercise:

• The expression \(x >= z || (x + y >= z)\) evaluates to true if either \(x\) is greater than or equal to \(z\), or their sum \(x + y\) is greater than or equal to \(z\).
• Such evaluations help in deciding the truthiness of conditions you may encounter in code.

Boolean expressions are a backbone for conditional logic in programming, where they help dictate true or false paths based on data-driven evaluations.