"It rains" Canonical CNF (CCNF) The differences between Contrapositive and Converse statements are tabulated below. Let us understand the terms "hypothesis" and "conclusion.". To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement. That means, any of these statements could be mathematically incorrect. Which of the other statements have to be true as well? Now it is time to look at the other indirect proof proof by contradiction. two minutes Q G Still wondering if CalcWorkshop is right for you? The original statement is true. The following theorem gives two important logical equivalencies. exercise 3.4.6. The contrapositive version of this theorem is "If x and y are two integers with opposite parity, then their sum must be odd." So we assume x and y have opposite parity. Tautology check Mixing up a conditional and its converse. If a quadrilateral is not a rectangle, then it does not have two pairs of parallel sides. Since a conditional statement and its contrapositive are logically equivalent, we can use this to our advantage when we are proving mathematical theorems. Solution. A statement obtained by reversing the hypothesis and conclusion of a conditional statement is called a converse statement. If a number is a multiple of 4, then the number is a multiple of 8. Proof Warning 2.3. Find the converse, inverse, and contrapositive of conditional statements. Your Mobile number and Email id will not be published. The Contrapositive of a Conditional Statement Suppose you have the conditional statement {\color {blue}p} \to {\color {red}q} p q, we compose the contrapositive statement by interchanging the hypothesis and conclusion of the inverse of the same conditional statement. (2020, August 27). When the statement P is true, the statement not P is false. https://www.thoughtco.com/converse-contrapositive-and-inverse-3126458 (accessed March 4, 2023). on syntax. paradox? The conditional statement is logically equivalent to its contrapositive. See more. In Preview Activity 2.2.1, we introduced the concept of logically equivalent expressions and the notation X Y to indicate that statements X and Y are logically equivalent. A statement that conveys the opposite meaning of a statement is called its negation. This means our contrapositive is : -q -p = "if n is odd then n is odd" We must prove or show the contraposition, that if n is odd then n is odd, if we can prove this to be true then we have. This is aconditional statement. A statement that is of the form "If p then q" is a conditional statement. } } } Emily's dad watches a movie if he has time. Graphical Begriffsschrift notation (Frege) Only two of these four statements are true! (Example #1a-e), Determine the logical conclusion to make the argument valid (Example #2a-e), Write the argument form and determine its validity (Example #3a-f), Rules of Inference for Quantified Statement, Determine if the quantified argument is valid (Example #4a-d), Given the predicates and domain, choose all valid arguments (Examples #5-6), Construct a valid argument using the inference rules (Example #7). Find the converse, inverse, and contrapositive. Let's look at some examples. To save time, I have combined all the truth tables of a conditional statement, and its converse, inverse, and contrapositive into a single table. If the converse is true, then the inverse is also logically true. Well, as we learned in our previous lesson, a direct proof always assumes the hypothesis is true and then logically deduces the conclusion (i.e., if p is true, then q is true). Okay, so a proof by contraposition, which is sometimes called a proof by contrapositive, flips the script. Figure out mathematic question. Elementary Foundations: An Introduction to Topics in Discrete Mathematics (Sylvestre), { "2.01:_Equivalence" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "2.02:_Propositional_Calculus" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "2.03:_Converse_Inverse_and_Contrapositive" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "2.04:_Activities" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "2.05:_Exercises" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Symbolic_language" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Logical_equivalence" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Boolean_algebra" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Predicate_logic" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Arguments" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Definitions_and_proof_methods" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Proof_by_mathematical_induction" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Axiomatic_systems" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Sets" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:_Recurrence_and_induction" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12:_Cardinality" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "13:_Countable_and_uncountable_sets" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "14:_Graphs" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15:_Paths_and_connectedness" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "16:_Trees_and_searches" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "17:_Relations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "18:_Equivalence_relations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "19:_Partially_ordered_sets" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "20:_Counting" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "21:_Permutations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "22:_Combinations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "23:_Binomial_and_multinomial_coefficients" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, 2.3: Converse, Inverse, and Contrapositive, [ "article:topic", "showtoc:no", "license:gnufdl", "Modus tollens", "authorname:jsylvestre", "licenseversion:13", "source@https://sites.ualberta.ca/~jsylvest/books/EF/book-elementary-foundations.html" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FCombinatorics_and_Discrete_Mathematics%2FElementary_Foundations%253A_An_Introduction_to_Topics_in_Discrete_Mathematics_(Sylvestre)%2F02%253A_Logical_equivalence%2F2.03%253A_Converse_Inverse_and_Contrapositive, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), source@https://sites.ualberta.ca/~jsylvest/books/EF/book-elementary-foundations.html, status page at https://status.libretexts.org. A conditional and its contrapositive are equivalent. So instead of writing not P we can write ~P. The inverse of The statement The right triangle is equilateral has negation The right triangle is not equilateral. The negation of 10 is an even number is the statement 10 is not an even number. Of course, for this last example, we could use the definition of an odd number and instead say that 10 is an odd number. We note that the truth of a statement is the opposite of that of the negation. Learn how to find the converse, inverse, contrapositive, and biconditional given a conditional statement in this free math video tutorial by Mario's Math Tutoring. A converse statement is the opposite of a conditional statement. Applies commutative law, distributive law, dominant (null, annulment) law, identity law, negation law, double negation (involution) law, idempotent law, complement law, absorption law, redundancy law, de Morgan's theorem. A statement formed by interchanging the hypothesis and conclusion of a statement is its converse. If you study well then you will pass the exam. Write the contrapositive and converse of the statement. - Converse of Conditional statement. All these statements may or may not be true in all the cases. (Examples #1-2), Express each statement using logical connectives and determine the truth of each implication (Examples #3-4), Finding the converse, inverse, and contrapositive (Example #5), Write the implication, converse, inverse and contrapositive (Example #6). If it rains, then they cancel school Example #1 It may sound confusing, but it's quite straightforward. We start with the conditional statement If Q then P. (Problem #1), Determine the truth value of the given statements (Problem #2), Convert each statement into symbols (Problem #3), Express the following in words (Problem #4), Write the converse and contrapositive of each of the following (Problem #5), Decide whether each of following arguments are valid (Problem #6, Negate the following statements (Problem #7), Create a truth table for each (Problem #8), Use a truth table to show equivalence (Problem #9). A function can only have an inverse if it is one-to-one so that no two elements in the domain are matched to the same element in the range. Converse, Inverse, and Contrapositive: Lesson (Basic Geometry Concepts) Example 2.12. Remember, we know from our study of equivalence that the conditional statement of if p then q has the same truth value of if not q then not p. Therefore, a proof by contraposition says, lets assume not q is true and lets prove not p. And consequently, if we can show not q then not p to be true, then the statement if p then q must be true also as noted by the State University of New York. The contrapositive of "If it rains, then they cancel school" is "If they do not cancel school, then it does not rain." If the statement is true, then the contrapositive is also logically true. ", "If John has time, then he works out in the gym. Do my homework now . -Inverse of conditional statement. Thus, the inverse is the implication ~\color{blue}p \to ~\color{red}q. }\) The contrapositive of this new conditional is \(\neg \neg q \rightarrow \neg \neg p\text{,}\) which is equivalent to \(q \rightarrow p\) by double negation. For example,"If Cliff is thirsty, then she drinks water." In the above example, since the hypothesis and conclusion are equivalent, all four statements are true. Contrapositive Formula ThoughtCo, Aug. 27, 2020, thoughtco.com/converse-contrapositive-and-inverse-3126458. The contrapositive of a conditional statement is a combination of the converse and the inverse. The truth table for Contrapositive of the conditional statement If p, then q is given below: Similarly, the truth table for the converse of the conditional statement If p, then q is given as: For more concepts related to mathematical reasoning, visit byjus.com today! Because a biconditional statement p q is equivalent to ( p q) ( q p), we may think of it as a conditional statement combined with its converse: if p, then q and if q, then p. The double-headed arrow shows that the conditional statement goes . Contradiction? An inversestatement changes the "if p then q" statement to the form of "if not p then not q. U - Contrapositive of a conditional statement. Hope you enjoyed learning! It is also called an implication. English words "not", "and" and "or" will be accepted, too. Suppose you have the conditional statement {\color{blue}p} \to {\color{red}q}, we compose the contrapositive statement by interchanging the hypothesis and conclusion of the inverse of the same conditional statement. The assertion A B is true when A is true (or B is true), but it is false when A and B are both false. To calculate the inverse of a function, swap the x and y variables then solve for y in terms of x. with Examples #1-9. is enabled in your browser. (If not p, then not q), Contrapositive statement is "If you did not get a prize then you did not win the race." The converse is logically equivalent to the inverse of the original conditional statement. First, form the inverse statement, then interchange the hypothesis and the conclusion to write the conditional statements contrapositive. The converse statement is " If Cliff drinks water then she is thirsty". Together, we will work through countless examples of proofs by contrapositive and contradiction, including showing that the square root of 2 is irrational! 1: Modus Tollens A conditional and its contrapositive are equivalent. Operating the Logic server currently costs about 113.88 per year "&" (conjunction), "" or the lower-case letter "v" (disjunction), "" or Help If the statement is true, then the contrapositive is also logically true. The contrapositive statement is a combination of the previous two. The converse statement is "If Cliff drinks water, then she is thirsty.". (Examples #1-3), Equivalence Laws for Conditional and Biconditional Statements, Use De Morgans Laws to find the negation (Example #4), Provide the logical equivalence for the statement (Examples #5-8), Show that each conditional statement is a tautology (Examples #9-11), Use a truth table to show logical equivalence (Examples #12-14), What is predicate logic? Taylor, Courtney. -Inverse statement, If I am not waking up late, then it is not a holiday. If \(f\) is not continuous, then it is not differentiable. Given a conditional statement, we can create related sentences namely: converse, inverse, and contrapositive. That's it! If a quadrilateral does not have two pairs of parallel sides, then it is not a rectangle. A statement obtained by negating the hypothesis and conclusion of a conditional statement. The inverse of the given statement is obtained by taking the negation of components of the statement. A conditional statement is a statement in the form of "if p then q,"where 'p' and 'q' are called a hypothesis and conclusion. The symbol ~\color{blue}p is read as not p while ~\color{red}q is read as not q . A proof by contrapositive would look like: Proof: We'll prove the contrapositive of this statement . var vidDefer = document.getElementsByTagName('iframe'); Now we can define the converse, the contrapositive and the inverse of a conditional statement. What is also important are statements that are related to the original conditional statement by changing the position of P, Q and the negation of a statement. Here 'p' is the hypothesis and 'q' is the conclusion. Proof Corollary 2.3. The addition of the word not is done so that it changes the truth status of the statement. If a number is a multiple of 8, then the number is a multiple of 4. For example, the contrapositive of "If it is raining then the grass is wet" is "If the grass is not wet then it is not raining." Note: As in the example, the contrapositive of any true proposition is also true. If \(m\) is not an odd number, then it is not a prime number. Write the converse, inverse, and contrapositive statement of the following conditional statement. Solution. Suppose we start with the conditional statement If it rained last night, then the sidewalk is wet.. represents the negation or inverse statement. If a quadrilateral has two pairs of parallel sides, then it is a rectangle. - Conditional statement, If you do not read books, then you will not gain knowledge. For. There . I'm not sure what the question is, but I'll try to answer it. What is Quantification? It turns out that even though the converse and inverse are not logically equivalent to the original conditional statement, they are logically equivalent to one another. Notice that by using contraposition, we could use one of our basic definitions, namely the definition of even integers, to help us prove our claim, which, once again, made our job so much easier. So change org. Math Homework. You can find out more about our use, change your default settings, and withdraw your consent at any time with effect for the future by visiting Cookies Settings, which can also be found in the footer of the site. Sometimes you may encounter (from other textbooks or resources) the words antecedent for the hypothesis and consequent for the conclusion. A conditional statement defines that if the hypothesis is true then the conclusion is true. A \rightarrow B. is logically equivalent to. Yes! ", To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. The inverse statement given is "If there is no accomodation in the hotel, then we are not going on a vacation. Retrieved from https://www.thoughtco.com/converse-contrapositive-and-inverse-3126458. Corollary \(\PageIndex{1}\): Modus Tollens for Inverse and Converse. What is a Tautology? 2) Assume that the opposite or negation of the original statement is true. Write the converse, inverse, and contrapositive statement for the following conditional statement. Write a biconditional statement and determine the truth value (Example #7-8), Construct a truth table for each compound, conditional statement (Examples #9-12), Create a truth table for each (Examples #13-15). What Are the Converse, Contrapositive, and Inverse? Thats exactly what youre going to learn in todays discrete lecture. Taylor, Courtney. window.onload = init; 2023 Calcworkshop LLC / Privacy Policy / Terms of Service. is the hypothesis. The inverse If it did not rain last night, then the sidewalk is not wet is not necessarily true. In mathematics, we observe many statements with if-then frequently. "->" (conditional), and "" or "<->" (biconditional). Instead, it suffices to show that all the alternatives are false. Therefore. Optimize expression (symbolically) Be it worksheets, online classes, doubt sessions, or any other form of relation, its the logical thinking and smart learning approach that we, at Cuemath, believe in. To get the inverse of a conditional statement, we negate both thehypothesis and conclusion. and How do we write them? Graphical expression tree The contrapositive does always have the same truth value as the conditional. What is Symbolic Logic? Thus, we can relate the contrapositive, converse and inverse statements in such a way that the contrapositive is the inverse of a converse statement. It will help to look at an example. An example will help to make sense of this new terminology and notation. Now I want to draw your attention to the critical word or in the claim above. \(\displaystyle \neg p \rightarrow \neg q\), \(\displaystyle \neg q \rightarrow \neg p\). Contrapositive. The sidewalk could be wet for other reasons. C Contrapositive can be used as a strong tool for proving mathematical theorems because contrapositive of a statement always has the same truth table. To calculate the inverse of a function, swap the x and y variables then solve for y in terms of x. Because trying to prove an or statement is extremely tricky, therefore, when we use contraposition, we negate the or statement and apply De Morgans law, which turns the or into an and which made our proof-job easier! Graphical alpha tree (Peirce) What are common connectives? You may come across different types of statements in mathematical reasoning where some are mathematically acceptable statements and some are not acceptable mathematically. four minutes The If part or p is replaced with the then part or q and the If there is no accomodation in the hotel, then we are not going on a vacation. Courtney K. Taylor, Ph.D., is a professor of mathematics at Anderson University and the author of "An Introduction to Abstract Algebra.". (if not q then not p). vidDefer[i].setAttribute('src',vidDefer[i].getAttribute('data-src')); Converse, Inverse, and Contrapositive of Conditional Statement Suppose you have the conditional statement p q {\color{blue}p} \to {\color{red}q} pq, we compose the contrapositive statement by interchanging the. Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step Write the converse, inverse, and contrapositive statements and verify their truthfulness. Textual expression tree Given statement is -If you study well then you will pass the exam. Learning objective: prove an implication by showing the contrapositive is true. "If we have to to travel for a long distance, then we have to take a taxi" is a conditional statement. A rewording of the contrapositive given states the following: G has matching M' that is not a maximum matching of G iff there exists an M-augmenting path. A careful look at the above example reveals something. How to Use 'If and Only If' in Mathematics, How to Prove the Complement Rule in Probability, What 'Fail to Reject' Means in a Hypothesis Test, Definitions of Defamation of Character, Libel, and Slander, converse and inverse are not logically equivalent to the original conditional statement, B.A., Mathematics, Physics, and Chemistry, Anderson University, The converse of the conditional statement is If, The contrapositive of the conditional statement is If not, The inverse of the conditional statement is If not, The converse of the conditional statement is If the sidewalk is wet, then it rained last night., The contrapositive of the conditional statement is If the sidewalk is not wet, then it did not rain last night., The inverse of the conditional statement is If it did not rain last night, then the sidewalk is not wet.. This is the beauty of the proof of contradiction. As you can see, its much easier to assume that something does equal a specific value than trying to show that it doesnt. There can be three related logical statements for a conditional statement. So for this I began assuming that: n = 2 k + 1. FlexBooks 2.0 CK-12 Basic Geometry Concepts Converse, Inverse, and Contrapositive. For instance, If it rains, then they cancel school. There is an easy explanation for this. The steps for proof by contradiction are as follows: It may sound confusing, but its quite straightforward. Atomic negations Supports all basic logic operators: negation (complement), and (conjunction), or (disjunction), nand (Sheffer stroke), nor (Peirce's arrow), xor (exclusive disjunction), implication, converse of implication, nonimplication (abjunction), converse nonimplication, xnor (exclusive nor, equivalence, biconditional), tautology (T), and contradiction (F). S The calculator will try to simplify/minify the given boolean expression, with steps when possible. The conditional statement given is "If you win the race then you will get a prize.". If \(f\) is not differentiable, then it is not continuous. In other words, contrapositive statements can be obtained by adding not to both component statements and changing the order for the given conditional statements. Conjunctive normal form (CNF) It will also find the disjunctive normal form (DNF), conjunctive normal form (CNF), and negation normal form (NNF). Note that an implication and it contrapositive are logically equivalent. ( Apply de Morgan's theorem $$$\overline{X \cdot Y} = \overline{X} + \overline{Y}$$$ with $$$X = \overline{A} + B$$$ and $$$Y = \overline{B} + C$$$: Apply de Morgan's theorem $$$\overline{X + Y} = \overline{X} \cdot \overline{Y}$$$ with $$$X = \overline{A}$$$ and $$$Y = B$$$: Apply the double negation (involution) law $$$\overline{\overline{X}} = X$$$ with $$$X = A$$$: Apply de Morgan's theorem $$$\overline{X + Y} = \overline{X} \cdot \overline{Y}$$$ with $$$X = \overline{B}$$$ and $$$Y = C$$$: Apply the double negation (involution) law $$$\overline{\overline{X}} = X$$$ with $$$X = B$$$: $$$\overline{\left(\overline{A} + B\right) \cdot \left(\overline{B} + C\right)} = \left(A \cdot \overline{B}\right) + \left(B \cdot \overline{C}\right)$$$. Negations are commonly denoted with a tilde ~. From the given inverse statement, write down its conditional and contrapositive statements. This page titled 2.3: Converse, Inverse, and Contrapositive is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jeremy Sylvestre via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. three minutes - Conditional statement If it is not a holiday, then I will not wake up late.