Symbolic translation shows how implication appears in ordinary language. Conditional statements and programming decisions show the same logic in action. A program often tests a condition and performs an instruction depending on whether the condition is true or false. In this lesson, conditional statements and programming decisions will connect $p\to q$ with If-Then and If-Then-Else structures. :::definition[Conditional Statement] A conditional statement is an implication of the form $p\to q$. It asserts that if the hypothesis $p$ is true, then the conclusion $q$ follows. ::: :::example[Conditional Promise] Let $$\begin{align*} p &: \text{I score at least }80\text{ marks.}\\ q &: \text{I receive a certificate.} \end{align*}$$ The conditional statement is $p\to q$. In words, it says: If I score at least $80$ marks, then I receive a certificate. ::: For the conditional promise $p\to q$, there are four possible cases. If $p=1$ and $q=1$, the promise is kept, so $p\to q=1$. If $p=1$ and $q=0$, the promise is broken, so $p\to q=0$. If $p=0$ and $q=1$, the condition was not met but the certificate was still received, so $p\to q=1$. If $p=0$ and $q=0$, the condition was not met and the certificate was not received, so $p\to q=1$. :::definition[If-Then Decision Structure] In computer programming, an If-Then decision structure has the logical form If $p$ then $q$. The instruction $q$ is executed only when $p$ is true. If $p$ is false, then the program continues to the next instruction. ::: :::definition[Logical Form of If-Then] An If-Then decision structure has the symbolic form $p\to q$, where $p$ is a condition and $q$ is the instruction performed when $p$ is true. ::: :::definition[If-Then-Else Decision Structure] An If-Then-Else decision structure has the form If $p$ then $q$ else $r$. The instruction $q$ is executed when $p$ is true, and the instruction $r$ is executed when $p$ is false. ::: Use the branch explorer to connect the logical value of a condition with the action a program performs. Set the condition $p$ to true or false and observe whether the then-branch or else-branch is selected. The important point is that an If-Then-Else structure always chooses exactly one of the two listed actions. This visual model supports the symbolic cases used in programming decisions. :::scientific-preview[If Then Else Branch Explorer] ::: :::definition[Logical Form of If-Then-Else] An If-Then-Else decision structure is represented by $$\begin{equation*} \begin{cases} q, & \text{if }p\text{ is true},\\ r, & \text{if }p\text{ is false}. \end{cases} \end{equation*}$$ ::: :::example[Programming Interpretation] A condition such as $n\geq 10$ becomes a logical statement after a value is assigned to $n$. If $n=15$, then $n\geq 10$ is true. If $n=4$, then $n\geq 10$ is false. Thus, relational expressions in computer programs can be treated as logical statements after input values are known. ::: :::example[Ordinary Conditional Language] Let $$\begin{align*} s &: \text{The employee completes the training.}\\ t &: \text{The employee receives system access.} \end{align*}$$ The statement $s\to t$ means: If the employee completes the training, then the employee receives system access. The statement $t\to s$ means: The employee receives system access only if the employee completes the training. These two implications are not logically identical. ::: :::definition[Intended Biconditional in Ordinary Language] In ordinary language, a speaker may intend $s\leftrightarrow t$ even when only one implication is stated. In mathematical logic, the implication $p\to q$ and the biconditional $p\leftrightarrow q$ must be treated as different statements. Thus, a biconditional should be written explicitly when equivalence is intended. ::: :::solved-problem The loop terminates when $((a>0)\land(b<5))\lor\neg((c=2)\land(d>10))$ is true. Determine whether the loop terminates for: (1) $a=3,b=4,c=2,d=12$. (2) $a=0,b=4,c=1,d=12$. (3) $a=5,b=8,c=2,d=15$. (4) $a=1,b=2,c=3,d=20$. ::: :::solution Let $$\begin{equation*} L=((a>0)\land(b<5))\lor\neg((c=2)\land(d>10)). \end{equation*}$$ [1] For $a=3,b=4,c=2,d=12$, the first part is true and the second negated part is false. Hence, $L=1$, so the loop terminates. [2] For $a=0,b=4,c=1,d=12$, the first part is false and $(c=2)\land(d>10)$ is false, so its negation is true. Hence, $L=1$, so the loop terminates. [3] For $a=5,b=8,c=2,d=15$, the first part is false and $(c=2)\land(d>10)$ is true, so its negation is false. Hence, $L=0$, so the loop does not terminate. [4] For $a=1,b=2,c=3,d=20$, the first part is true. Hence, $L=1$, so the loop terminates. ::: :::exercise A program says: If the password is correct, then access is granted; else access is denied. Let $p$ mean the password is correct, $q$ mean access is granted, and $r$ mean access is denied. State the action when (1) $p=1$ and (2) $p=0$. ::: :::answer (1) If $p=1$, the program executes $q$, so access is granted. (2) If $p=0$, the program executes $r$, so access is denied. ::: :::faq[Frequently Asked Questions] Q: Is an If-Then statement the same as an If-Then-Else statement? A: No. If-Then specifies what happens when the condition is true, while If-Then-Else also specifies what happens when the condition is false. Q: Why is $p\to q$ true when $p$ is false? A: The conditional promise is not violated unless $p$ is true and $q$ is false. Q: Are programming conditions open sentences? A: Before input values are assigned, they may behave like open sentences. After values are assigned, they become true or false statements. ::: :::call-to-action[Connect Logic with Decisions] subtitle: Practise evaluating conditional program conditions before constructing larger compound truth tables. button: Next Lesson | /discrete-mathematics/mathematical-logic/basic-logical-connectives-and-truth-tables/parentheses-in-compound-truth-tables button-ghost: Previous Lesson | /discrete-mathematics/mathematical-logic/basic-logical-connectives-and-truth-tables/symbolic-translation-of-compound-statements :::