Contrapositive is used when an implication has many hypotheses or when the hypothesis specifies infinitely many objects. If n > 2, then n 2 > 4. 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. Before we define the converse, contrapositive, and inverse of a conditional statement, we need to examine the topic of negation. Thats exactly what youre going to learn in todays discrete lecture. The conditional statement given is "If you win the race then you will get a prize.". Together, we will work through countless examples of proofs by contrapositive and contradiction, including showing that the square root of 2 is irrational! "If Cliff is thirsty, then she drinks water"is a condition. An indirect proof doesnt require us to prove the conclusion to be true. ," we can create three related statements: A conditional statement consists of two parts, a hypothesis in the if clause and a conclusion in the then clause. 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). There . Dont worry, they mean the same thing. These are the two, and only two, definitive relationships that we can be sure of. Then show that this assumption is a contradiction, thus proving the original statement to be true. Thus. Thus, there are integers k and m for which x = 2k and y . If you eat a lot of vegetables, then you will be healthy. 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. Taylor, Courtney. There are 3 methods for finding the inverse of a function: algebraic method, graphical method, and numerical method. ", To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. } } } For example,"If Cliff is thirsty, then she drinks water." Contradiction? The contrapositive If the sidewalk is not wet, then it did not rain last night is a true statement. 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. two minutes - Contrapositive of a conditional statement. The inverse of the given statement is obtained by taking the negation of components of the statement. E Use Venn diagrams to determine if the categorical syllogism is valid or invalid (Examples #1-4), Determine if the categorical syllogism is valid or invalid and diagram the argument (Examples #5-8), Identify if the proposition is valid (Examples #9-12), Which of the following is a proposition? (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? - Contrapositive statement. Write the contrapositive and converse of the statement. When the statement P is true, the statement not P is false. ", The inverse statement is "If John does not have time, then he does not work out in the gym.". If a quadrilateral is a rectangle, then it has two pairs of parallel sides. - Converse of Conditional statement. Yes! Apply this result to show that 42 is irrational, using the assumption that 2 is irrational. Take a Tour and find out how a membership can take the struggle out of learning math. Suppose if p, then q is the given conditional statement if q, then p is its contrapositive statement. Cookies collect information about your preferences and your devices and are used to make the site work as you expect it to, to understand how you interact with the site, and to show advertisements that are targeted to your interests. \(\displaystyle \neg p \rightarrow \neg q\), \(\displaystyle \neg q \rightarrow \neg p\). 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For. Graphical alpha tree (Peirce) Accessibility StatementFor more information contact us [email protected] check out our status page at https://status.libretexts.org. Here 'p' is the hypothesis and 'q' is the conclusion. is The original statement is true. A converse statement is the opposite of a conditional statement. If the conditional is true then the contrapositive is true. Note that an implication and it contrapositive are logically equivalent. The sidewalk could be wet for other reasons. (2020, August 27). 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}. On the other hand, the conclusion of the conditional statement \large{\color{red}p} becomes the hypothesis of the converse. This version is sometimes called the contrapositive of the original conditional statement. Instead of assuming the hypothesis to be true and the proving that the conclusion is also true, we instead, assumes that the conclusion to be false and prove that the hypothesis is also false. five minutes If you read books, then you will gain knowledge. If a number is a multiple of 8, then the number is a multiple of 4. Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step Required fields are marked *. 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. Solution. You may use all other letters of the English A contradiction is an assertion of Propositional Logic that is false in all situations; that is, it is false for all possible values of its variables. The converse statement is " If Cliff drinks water then she is thirsty". Example Step 3:. Therefore, the converse is the implication {\color{red}q} \to {\color{blue}p}. - Conditional statement, If you are healthy, then you eat a lot of vegetables. Contrapositive definition, of or relating to contraposition. "->" (conditional), and "" or "<->" (biconditional). Which of the other statements have to be true as well? https://www.thoughtco.com/converse-contrapositive-and-inverse-3126458 (accessed March 4, 2023). They are related sentences because they are all based on the original conditional statement. AtCuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students! 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! The converse and inverse may or may not be true. In mathematics, we observe many statements with if-then frequently. It is to be noted that not always the converse of a conditional statement is true. Prove that if x is rational, and y is irrational, then xy is irrational. Textual expression tree The contrapositive of a conditional statement is a combination of the converse and the inverse. (If not q then not p). A conditional statement takes the form If p, then q where p is the hypothesis while q is the conclusion. function init() { Write the contrapositive and converse of the statement. Related to the conditional \(p \rightarrow q\) are three important variations. A non-one-to-one function is not invertible. 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 $$$X = B$$$: $$$\overline{\left(\overline{A} + B\right) \cdot \left(\overline{B} + C\right)} = \left(A \cdot \overline{B}\right) + \left(B \cdot \overline{C}\right)$$$. A statement obtained by reversing the hypothesis and conclusion of a conditional statement is called a converse statement. Solution We use the contrapositive that states that function f is a one to one function if the following is true: if f(x 1) = f(x 2) then x 1 = x 2 We start with f(x 1) = f(x 2) which gives a x 1 + b = a x 2 + b Simplify to obtain a ( x 1 - x 2) = 0 Since a 0 the only condition for the above to be satisfied is to have x 1 - x 2 = 0 which . Contrapositive Formula If it rains, then they cancel school Optimize expression (symbolically and semantically - slow) contrapositive of the claim and see whether that version seems easier to prove. 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. The converse is logically equivalent to the inverse of the original conditional statement. (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).

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