Question

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

Short answer

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

Step-by-step solution

Evaluate Expression a

Begin by evaluating the expression $!(x>10)$. Since $x=10$, check if $x>10$ is true. Here, $10$ is not greater than $10$, so $x>10$ is false. The negation operator $!)$ then makes the entire expression true.

Evaluate Expression b

Evaluate the expression $x<=5||y<15$. For $x<=5$, since $x=10$, this is false. Then, evaluate $y<15$. Since $y=15$, it's not less than 15, making this part false. Using the $||$ operator, which needs at least one true condition to be true, the entire expression is false.

Evaluate Expression c

Evaluate the expression . Start with $x!=5$, which is true since $10$ is not equal to $5$. Next, check $y!=z$, $15$ is not equal to $20$, making this true. Both conditions are true, so the expression is true.

Evaluate Expression d

Evaluate the expression $x>=z||(x+y>=z)$. Checking $x>=z$, since $10$ is not greater than or equal to $20$, it is false. For $x+y>=z$, substitute values to get $10+15=25$, which is greater than $20$. Since one part is true, the expression evaluates to true using the $||$ operator.

Evaluate Expression e

Evaluate the expression . Begin with $x<=y-2$, since $10$ is less than $13$, this is true. Check $y>=z$, which is false as $15$ is not greater than or equal to $20$. Next, evaluate $(z-2!=20)$. Here, $z-2=18$, which is not equal to $20$, making it true. The expression requires one true statement due to $||$, so it evaluates to true.

Key concepts

Logical Operators

Logical operators are fundamental tools used in programming to perform logical operations on expressions. The common logical operators are AND (&&), OR (||), and NOT (!). They allow us to build complex conditions in our code by combining multiple logical statements.

• The AND (&&) operator evaluates to true if both expressions on either side are true. Otherwise, it results in false. For example, if we have $(x>5)$ and $(y<10)$, our combined condition $(x>5)\mathrm{\&}\mathrm{\&}(y<10)$ is true only if both conditions are true.
• The OR (||) operator evaluates to true if at least one of the expressions is true. So, for conditions $(a==2)$ and $(b==5)$, the expression $(a==2)||(b==5)$ is true if either or both conditions are true.
• The NOT (!) operator inverts the boolean value, making false conditions true and true conditions false. For instance, $!(x>0)$ is true only when $x$ is not greater than 0.

Variable Comparison

Variable comparison involves evaluating relationships between different variables using relational operators such as greater than (>), less than (<), equal to (==), not equal to (!=), and so on. These comparisons are essential for decision-making in programs.

• For example, the expression $x>10$ checks if variable $x$ is greater than 10, returning true or false accordingly.
• The equality check $x==y$ evaluates to true if $x$ holds the same value as $y$, while $x!=y$ checks if $x$ and $y$ differ.
• Comparisons such as $x>=z$ or $y<=z$ include equality in their relation, giving wider conditions to evaluate as true.

These operators are crucial in making logical decisions based on variable values.

Conditional Statements

Conditional statements use logical operators and variable comparisons to make decisions within code. They typically follow the "if-else" structure, allowing specific blocks of code to execute based on evaluated conditions.

• An $if$ statement runs a code block if the condition is true. For instance, $if(x>10)$ {} checks if $x$ is greater than 10 and executes the code block if it is.
• The $else$ statement pairs with $if$, executing an alternative block of code if the initial condition is false.
• Additionally, $elseif$ can be used for multiple conditions, checking each sequentially and executing the matching block of code.

These constructs are key to controlling the flow of a program, reacting dynamically to different input values or states.

Expression Evaluation

Expression evaluation refers to the process of determining the truth value (true or false) of an expression, which often includes logical operators and variable comparisons. This step is essential in executing conditional logic based on results.

• Begin by evaluating each part of a composite expression individually. For instance, in $a>b||c<d$, determine the truth of $a>b$ and $c<d$ separately before applying the OR logic.
• Logical operators then combine these results. With the OR (||) operator in this example, only one need be true for the entire expression to evaluate to true.
• Sometimes expressions involve arithmetic or more complex combinations of logical and relational operations, requiring careful order of operations to ensure correct evaluation.

Through expression evaluation, programs make informed decisions: selecting paths, validating rules, and generally steering the course of the execution based on given conditions.