Determine if each resulting statement is true or false. 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. You don't know anything if I . three minutes Prove that if x is rational, and y is irrational, then xy is irrational. Contrapositive proofs work because if the contrapositive is true, due to logical equivalence, the original conditional statement is also true. This is the beauty of the proof of contradiction. Get access to all the courses and over 450 HD videos with your subscription. In addition, the statement If p, then q is commonly written as the statement p implies q which is expressed symbolically as {\color{blue}p} \to {\color{red}q}. 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. 6. The symbol ~\color{blue}p is read as not p while ~\color{red}q is read as not q . Contrapositive Formula The contrapositive of a statement negates the hypothesis and the conclusion, while swaping the order of the hypothesis and the conclusion. That's it! Legal. If it is false, find a counterexample. -Inverse of conditional statement. "It rains" Therefore, the converse is the implication {\color{red}q} \to {\color{blue}p}. And then the country positive would be to the universe and the convert the same time. Thats exactly what youre going to learn in todays discrete lecture. Prove by contrapositive: if x is irrational, then x is irrational. The converse statement is " If Cliff drinks water then she is thirsty". Notice, the hypothesis \large{\color{blue}p} of the conditional statement becomes the conclusion of the converse. Prove the proposition, Wait at most The converse statements are formed by interchanging the hypothesis and conclusion of given conditional statements. "->" (conditional), and "" or "<->" (biconditional). preferred. How do we show propositional Equivalence? 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. We may wonder why it is important to form these other conditional statements from our initial one. The contrapositive of a conditional statement is a combination of the converse and the inverse. is The original statement is the one you want to prove. ) Then show that this assumption is a contradiction, thus proving the original statement to be true. (Example #18), Construct a truth table for each statement (Examples #19-20), Create a truth table for each proposition (Examples #21-24), Form a truth table for the following statement (Example #25), What are conditional statements? So for this I began assuming that: n = 2 k + 1. Corollary \(\PageIndex{1}\): Modus Tollens for Inverse and Converse. (Examples #1-2), Understanding Universal and Existential Quantifiers, Transform each sentence using predicates, quantifiers and symbolic logic (Example #3), Determine the truth value for each quantified statement (Examples #4-12), How to Negate Quantified Statements? with Examples #1-9. ThoughtCo, Aug. 27, 2020, thoughtco.com/converse-contrapositive-and-inverse-3126458. Quine-McCluskey optimization "If it rains, then they cancel school" What is Quantification? We can also construct a truth table for contrapositive and converse statement. represents the negation or inverse statement. var vidDefer = document.getElementsByTagName('iframe'); If \(m\) is an odd number, then it is a prime number. Optimize expression (symbolically and semantically - slow) (If not q then not p). The If \(f\) is differentiable, then it is continuous. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Write the contrapositive and converse of the statement. Taylor, Courtney. There are 3 methods for finding the inverse of a function: algebraic method, graphical method, and numerical method. 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. The most common patterns of reasoning are detachment and syllogism. There are 3 methods for finding the inverse of a function: algebraic method, graphical method, and numerical method. 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) What are the properties of biconditional statements and the six propositional logic sentences? Graphical Begriffsschrift notation (Frege) The converse statement for If a number n is even, then n2 is even is If a number n2 is even, then n is even. Solution. Simplify the boolean expression $$$\overline{\left(\overline{A} + B\right) \cdot \left(\overline{B} + C\right)}$$$. What we see from this example (and what can be proved mathematically) is that a conditional statement has the same truth value as its contrapositive. Below is the basic process describing the approach of the proof by contradiction: 1) State that the original statement is false. If \(m\) is a prime number, then it is an odd number. Contrapositive Proof Even and Odd Integers. The mini-lesson targetedthe fascinating concept of converse statement. The contrapositive of this statement is If not P then not Q. Since the inverse is the contrapositive of the converse, the converse and inverse are logically equivalent. "They cancel school" The hypothesis 'p' and conclusion 'q' interchange their places in a converse statement. Jenn, Founder Calcworkshop, 15+ Years Experience (Licensed & Certified Teacher). Now you can easily find the converse, inverse, and contrapositive of any conditional statement you are given! Math Homework. What are the 3 methods for finding the inverse of a function? A non-one-to-one function is not invertible. Let x and y be real numbers such that x 0. Help Find the converse, inverse, and contrapositive of conditional statements. If the converse is true, then the inverse is also logically true. Dont worry, they mean the same thing. Given an if-then statement "if Like contraposition, we will assume the statement, if p then q to be false. Graphical expression tree In other words, contrapositive statements can be obtained by adding not to both component statements and changing the order for the given conditional statements. S If the conditional is true then the contrapositive is true. The original statement is true. Write the converse, inverse, and contrapositive statement of the following conditional statement. Proof Warning 2.3. For. A statement that conveys the opposite meaning of a statement is called its negation. "If Cliff is thirsty, then she drinks water"is a condition. To form the converse of the conditional statement, interchange the hypothesis and the conclusion. 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. 1. 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. Here are some of the important findings regarding the table above: Introduction to Truth Tables, Statements, and Logical Connectives, Truth Tables of Five (5) Common Logical Connectives or Operators. So if battery is not working, If batteries aren't good, if battery su preventing of it is not good, then calculator eyes that working. There are two forms of an indirect proof. We will examine this idea in a more abstract setting. ( To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement. For instance, If it rains, then they cancel school. Take a Tour and find out how a membership can take the struggle out of learning math. Lets look at some examples. 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 $$$X = B$$$: $$$\overline{\left(\overline{A} + B\right) \cdot \left(\overline{B} + C\right)} = \left(A \cdot \overline{B}\right) + \left(B \cdot \overline{C}\right)$$$. Select/Type your answer and click the "Check Answer" button to see the result. A pattern of reaoning is a true assumption if it always lead to a true conclusion. not B \rightarrow not A. If \(f\) is not continuous, then it is not differentiable. 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). (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? Textual alpha tree (Peirce) Determine if inclusive or or exclusive or is intended (Example #14), Translate the symbolic logic into English (Example #15), Convert the English sentence into symbolic logic (Example #16), Determine the truth value of each proposition (Example #17), How do we create a truth table? Whats the difference between a direct proof and an indirect proof? Taylor, Courtney. See more. The contrapositive of the conditional statement is "If not Q then not P." The inverse of the conditional statement is "If not P then not Q." if p q, p q, then, q p q p For example, If it is a holiday, then I will wake up late. An inversestatement changes the "if p then q" statement to the form of "if not p then not q. truth and falsehood and that the lower-case letter "v" denotes the The converse statement is "You will pass the exam if you study well" (if q then p), The inverse statement is "If you do not study well then you will not pass the exam" (if not p then not q), The contrapositive statement is "If you didnot pass the exam then you did notstudy well" (if not q then not p). In other words, the negation of p leads to a contradiction because if the negation of p is false, then it must true. Starting with an original statement, we end up with three new conditional statements that are named the converse, the contrapositive, and the inverse. exercise 3.4.6. Proof Corollary 2.3. Sometimes you may encounter (from other textbooks or resources) the words antecedent for the hypothesis and consequent for the conclusion. (If p then q), Contrapositive statement is "If we are not going on a vacation, then there is no accomodation in the hotel." "If they cancel school, then it rains. - Conditional statement If it is not a holiday, then I will not wake up late. That is to say, it is your desired result. Mathwords: Contrapositive Contrapositive Switching the hypothesis and conclusion of a conditional statement and negating both. enabled in your browser. 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. A \rightarrow B. is logically equivalent to. This video is part of a Discrete Math course taught at the University of Cinc. A proof by contrapositive would look like: Proof: We'll prove the contrapositive of this statement . Suppose that the original statement If it rained last night, then the sidewalk is wet is true. 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. -Conditional statement, If it is not a holiday, then I will not wake up late. For example,"If Cliff is thirsty, then she drinks water." If \(m\) is not a prime number, then it is not an odd number. If it rains, then they cancel school A statement which is of the form of "if p then q" is a conditional statement, where 'p' is called hypothesis and 'q' is called the conclusion. 10 seconds It is also called an implication. Suppose if p, then q is the given conditional statement if q, then p is its converse statement. paradox? ThoughtCo. 2023 Calcworkshop LLC / Privacy Policy / Terms of Service, What is a proposition? A statement obtained by exchangingthe hypothesis and conclusion of an inverse statement. 2) Assume that the opposite or negation of the original statement is true. Together, we will work through countless examples of proofs by contrapositive and contradiction, including showing that the square root of 2 is irrational! 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. Not to G then not w So if calculator. 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If two angles are not congruent, then they do not have the same measure. Then show that this assumption is a contradiction, thus proving the original statement to be true. The contrapositive statement for If a number n is even, then n2 is even is If n2 is not even, then n is not even. 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. Graphical alpha tree (Peirce) Contrapositive. What are common connectives? Required fields are marked *. Not every function has an inverse. If there is no accomodation in the hotel, then we are not going on a vacation. Write the contrapositive and converse of the statement. Example 1.6.2. Similarly, for all y in the domain of f^(-1), f(f^(-1)(y)) = y. }\) 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. P Yes! Heres a BIG hint. If two angles do not have the same measure, then they are not congruent. The converse If the sidewalk is wet, then it rained last night is not necessarily true. 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 . Here are a few activities for you to practice. Every statement in logic is either true or false. This version is sometimes called the contrapositive of the original conditional statement. Only two of these four statements are true! If a quadrilateral is not a rectangle, then it does not have two pairs of parallel sides. Learn from the best math teachers and top your exams, Live one on one classroom and doubt clearing, Practice worksheets in and after class for conceptual clarity, Personalized curriculum to keep up with school, 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, Interactive Questions on Converse Statement, if \(\begin{align} p \rightarrow q,\end{align}\) then, \(\begin{align} q \rightarrow p\end{align}\), if \(\begin{align} p \rightarrow q,\end{align}\) then, \(\begin{align} \sim{p} \rightarrow \sim{q}\end{align}\), if \(\begin{align} p \rightarrow q,\end{align}\) then, \(\begin{align} \sim{q} \rightarrow \sim{p}\end{align}\), if \(\begin{align} p \rightarrow q,\end{align}\) then, \(\begin{align} q \rightarrow p\end{align}\). The contrapositive does always have the same truth value as the conditional. If 2a + 3 < 10, then a = 3. They are related sentences because they are all based on the original conditional statement. Suppose we start with the conditional statement If it rained last night, then the sidewalk is wet.. 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.. Before we define the converse, contrapositive, and inverse of a conditional statement, we need to examine the topic of negation.