Conjunction and disjunction combine statements symmetrically, but implication introduces direction. The statement $p\to q$ says that the truth of $p$ leads to the truth of $q$. This direction is essential in definitions, theorems, proofs, algorithms, and everyday mathematical language. In this lesson, implication and biconditional will be studied through hypotheses, conclusions, necessary conditions, and sufficient conditions. :::definition[Implication] Let $p$ and $q$ be statements. The implication from $p$ to $q$ is denoted by $p\to q$ and is read as if $p$, then $q$. ::: :::definition[Hypothesis and Conclusion] In the implication $p\to q$, the statement $p$ is called the hypothesis, and the statement $q$ is called the conclusion. ::: :::definition[Sufficient Condition] In the implication $p\to q$, the statement $p$ is a sufficient condition for $q$. This means that if $p$ is true, then $q$ must be true. ::: :::definition[Necessary Condition] In the implication $p\to q$, the statement $q$ is a necessary condition for $p$. This means that $p$ cannot be true unless $q$ is also true. ::: The implication $p\to q$ may be expressed in several equivalent ways: If $p$, then $q$; $p$ is sufficient for $q$; $q$ is necessary for $p$; and $p$ only if $q$. Students often reverse necessary and sufficient conditions. The phrase $p$ only if $q$ means $p\to q$, not $q\to p$. Use the translator to focus on the direction of an implication. Read each phrase carefully, decide which statement forces the other, and then choose the symbolic form. The phrase sufficient for usually points from the sufficient condition to the required conclusion. The phrase necessary for usually points toward the necessary condition, so this is where many reversal errors occur. :::calculator[Necessary and Sufficient Translator] ::: :::example[Implication] Let $$\begin{align*} p &: \text{A number is divisible by }4.\\ q &: \text{The number is even.} \end{align*}$$ Then $p\to q$ means If a number is divisible by $4$, then the number is even. Here divisibility by $4$ is sufficient for being even, and being even is necessary for divisibility by $4$. ::: :::definition[Truth Table for Implication] Let $p$ and $q$ be statements. The truth table for $p\to q$ is $$\begin{equation*} \begin{array}{c|c|c} p & q & p\to q\\ \hline 0 & 0 & 1\\ 0 & 1 & 1\\ 1 & 0 & 0\\ 1 & 1 & 1 \end{array} \end{equation*}$$ Thus, $p\to q$ is false only when $p$ is true and $q$ is false. ::: :::definition[Biconditional] Let $p$ and $q$ be statements. The biconditional of $p$ and $q$ is denoted by $p\leftrightarrow q$ and is read as $p$ if and only if $q$. ::: :::definition[Necessary and Sufficient Condition] In the biconditional $p\leftrightarrow q$, each statement is both necessary and sufficient for the other. Thus, $p$ and $q$ must have the same truth value. ::: :::definition[Truth Table for Biconditional] Let $p$ and $q$ be statements. The truth table for $p\leftrightarrow q$ is $$\begin{equation*} \begin{array}{c|c|c} p & q & p\leftrightarrow q\\ \hline 0 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0\\ 1 & 1 & 1 \end{array} \end{equation*}$$ Thus, $p\leftrightarrow q$ is true exactly when $p$ and $q$ have the same truth value. ::: :::example[Biconditional] Let $$\begin{align*} p &: \text{A triangle has three equal sides.}\\ q &: \text{A triangle is equilateral.} \end{align*}$$ Then $p\leftrightarrow q$ means A triangle has three equal sides if and only if it is equilateral. ::: :::solved-problem Let $p\to q$ be false. Determine the truth values of $p\land q$, $\neg p\lor q$, $q\to p$, and $\neg q\to\neg p$. ::: :::solution Let $p\to q$ be false. An implication is false only when the hypothesis is true and the conclusion is false. Therefore, $p=1$ and $q=0$. Hence, $$\begin{align*} p\land q &= 1\land 0=0,\\ \neg p\lor q &= 0\lor 0=0,\\ q\to p &= 0\to 1=1,\\ \neg q\to\neg p &= 1\to 0=0. \end{align*}$$ Thus, the truth values are $0$, $0$, $1$, and $0$ respectively. ::: :::solved-problem Determine the truth value of each implication: (1) If $2+5=10$, then $4+1=7$. (2) If $6$ is divisible by $3$, then $12$ is divisible by $4$. (3) If $8$ is even, then $15$ is even. (4) If Earth has one moon, then $3+3=6$. ::: :::solution [1] The hypothesis is false and the conclusion is false. Therefore, the implication is true. [2] The hypothesis is true and the conclusion is true. Therefore, the implication is true. [3] The hypothesis is true and the conclusion is false. Therefore, the implication is false. [4] The hypothesis is true and the conclusion is true. Therefore, the implication is true. Hence, the truth values are true, true, false, and true. ::: :::solved-problem Rewrite each statement in if-then form: (1) Regular attendance is sufficient for eligibility to take the final examination. (2) A password is required to access the online portal. (3) A student may borrow books only if the student has a valid library card. ::: :::solution [1] Since regular attendance is sufficient for eligibility, the if-then form is: If a student has regular attendance, then the student is eligible to take the final examination. [2] Since a password is required to access the online portal, access implies having a password. The if-then form is: If a person accesses the online portal, then the person has a password. [3] Since a student may borrow books only if the student has a valid library card, borrowing implies having a valid card. The if-then form is: If a student borrows books, then the student has a valid library card. ::: :::exercise Let $p:$ A student passes the course, and $q:$ The student completes all required assessments. Translate: (1) Completing all required assessments is necessary for passing the course. (2) Passing the course is sufficient for completing all required assessments. (3) The student passes the course if and only if the student completes all required assessments. ::: :::answer (1) $p\to q$. (2) $p\to q$. (3) $p\leftrightarrow q$. ::: :::faq[Frequently Asked Questions] Q: When is an implication false? A: The implication $p\to q$ is false only when $p$ is true and $q$ is false. Q: Does $p\to q$ mean the same as $q\to p$? A: No. The direction matters. The first has hypothesis $p$, while the second has hypothesis $q$. Q: What does if and only if mean? A: It means both implications hold: $p\to q$ and $q\to p$. ::: :::call-to-action[Review Conditional Language] subtitle: Practise necessary and sufficient conditions before studying open sentences and truth values. button: Next Lesson | /discrete-mathematics/mathematical-logic/basic-logical-connectives-and-truth-tables/open-sentences-and-truth-values button-ghost: Previous Lesson | /discrete-mathematics/mathematical-logic/basic-logical-connectives-and-truth-tables/conjunction-and-disjunction :::