Question

Assume the variables x = 5, y = 6, and z = 8. Indicate if each of the following conditions is true or false: A) (x >= 0) || (x <= y) B) (z - y) > y C) !((z - y) > x)

Short answer

Question: Based on the given values of x, y, and z, evaluate the following conditions and determine if they are true or false. x = 5, y = 6, and z = 8. Condition A: (x >= 0) || (x <= y) Condition B: (z - y) > y Condition C: !((z - y) > x) Answer: For the given values, Condition A is True, Condition B is False, and Condition C is True.

Step-by-step solution

Identify the variables and their values

x = 5, y = 6, and z = 8.

Evaluate the expression in Condition A

(x >= 0) || (x <= y) Replace x and y with their given values: (5 >= 0) || (5 <= 6) Both expressions are true, so the whole condition is true.

Evaluate the expression in Condition B

(z - y) > y Replace z and y with their given values: (8 - 6) > 6 2 > 6 The condition is false.

Evaluate the expression in Condition C

!((z - y) > x) Replace x, y, and z with their given values: !((8 - 6) > 5) !(2 > 5) The inner expression is false, so the whole condition with NOT is true. Final solutions: A) True B) False C) True

Key concepts

Logical Operators: The Building Blocks

Logical operators are crucial in programming and mathematics as they allow us to combine multiple conditions and expressions to make complex decisions. Three primary logical operators are often used: AND (\(&&\)), OR (\(||\)), and NOT (\(!\)). Each of these operators plays a unique role in evaluating expressions.

• AND (\(&&\)): This operator evaluates to true only if all the combined conditions are true.
• OR (\(||\)): This operator requires only one of the conditions to be true for the whole expression to evaluate as true.
• NOT (\(!\)): This operator simply flips the truth value of the condition it precedes. It turns true into false and vice versa.

In our example, Condition A, "(x >= 0) || (x <= y)", utilizes the OR operator, meaning it checks if at least one of the included conditions is true. In contrast, Condition C uses the NOT operator to reverse the result of the expression within the parentheses. These logical operators form the backbone of decision-making in programming contexts.

Conditional Statements: Making Decisions

Conditional statements are the way computers make decisions and determine which instructions to execute next. They allow programs to execute certain code blocks only when specific conditions are met.

In the context of the exercise, each condition acts as a simple example of this concept. Each statement is evaluated to decide if it is true or false, which then influences the flow of the program.

For Condition A, "(x >= 0) || (x <= y)", the presence of the OR operator means that if either x is greater than or equal to 0, or x is less than or equal to y, the statement will be true, allowing the program to proceed based on this truth.

Condition B, "(z - y) > y", represents an instance where a simple arithmetic evaluation is required before the logical condition is assessed. This condition determines if the left-hand side of the statement is greater than the right-hand side.
Lastly, Condition C, "!((z - y) > x)", involves the NOT operator, which inverts the simple evaluation, demonstrating more complex decision-making built on nesting expressions.

Variable Evaluation: Understanding and Substitution

Variable evaluation is a fundamental process that involves replacing variables with their respective values to evaluate an expression. By understanding and substituting variable values, we can solve, predict, and verify the outcome of expressions.

In the exercise, we first identify that x = 5, y = 6, and z = 8. We then substitute these values into each condition to determine whether the expression evaluates to true or false.

• In Condition A, "(x >= 0) || (x <= y)", we replace x and y with their values to get "(5 >= 0) || (5 <= 6)". Both parts are true, making the entire condition true due to the OR operator.
• In Condition B, "(z - y) > y", substituting z and y gives us "(8 - 6) > 6", simplifying to "2 > 6", which is false.
• Condition C, "!((z - y) > x)", after substitution becomes "!((8 - 6) > 5)" or !"(2 > 5)". Since "2 > 5" is false, the NOT operator flips it to true.

These simple evaluation and substitution steps are key for mathematical problem-solving and are commonly used in algorithms and programming.