Mathematical proofmethods of proofproof by contrapositive. Converses, contrapositives and proof by the contrapositive. Proof by contrapositive proof by contrapositive takes advantage of the logical equivalence between p implies q and not q implies not p. In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive. Learn contrapositive with free interactive flashcards. We started with direct proofs, and then we moved on to proofs by contradiction and mathematical induction. You can either select the files you want to merge from you computer or drop them on. When to use the contrapositive to prove a statment. Prove the following statement by contraposition if a product of two positive real numbers is greater than 100, then at least one of the number is greater than 10. A proof by contradiction is a method of proving a statement by assuming the hypothesis to be true and conclusion to be false, and then deriving a contradiction. Logic and proof converse, inverse, and contrapositive. By the closure property, we know b is an integer, so we see that 3jn2.
For example, the assertion if it is my car, then it is red is equivalent to if that car is not red, then it is not mine. The method of contradiction is an example of an indirect proof. You can prove this by looking at their truth tables. Choose from 220 different sets of contrapositive flashcards on quizlet. If three angles are equal in a triangle, then the triangle is an equilateral. In classical logic, an implication is logically equivalent to its contrapositive, and, moreover, its inverse is logically equivalent to its converse. So, to prove if p, then q by the method of contrapositive means to prove if. Proof by contraposition examples kent state university.
Contraposition definition of contraposition by merriamwebster. Notes on proof by contrapositive and proof by contradiction. Contrapositive definition of contrapositive by merriam. The contrapositive is certainly true because the entire province of bc is a part of canada.
Explicitly state which truth value goes to each statement. What are the converse, contrapositive, and inverse. For instance, if it rains, then they cancel school. You are probably familiar with at least a little canadian geography, and know that british columbia bc is a geographical area within canada.
Similarly, a statements converse and its inverse are always either both true or both false. Choose from a variety of file types multiple pdf files, microsoft word documents, microsoft excel spreadsheets, microsoft powerpoint. Contrapositive definition of contrapositive by the free. The contrapositive is formed by negating both the hypothesis and the conclusion, and then interchanging the resulting negations. Pdf merge combine pdf files free tool to merge pdf online. Therefore, the contrapositive of a definition is always true. D s is the contrapositive of s d just as much as s d is the contrapositive of d s. The negation of a statement simply involves the insertion of the word not at the proper part of the statement. Inverse converse and contrapositive statements flashcards. Converse, inverse, and contrapositive examples video. It just means that youre referring to the only other way in the lawgic language of expressing that particular conditional relationship. Contrapositive definition of contrapositive by merriamwebster. If q, then p that is, the contrapositive of hence, contrapositive of a statement is described as reversal and negation both the statements.
Formulas and equations also important info on slope. Some problems have structures that make it more obvious to try induction or contradiction. However, in a proof by contradiction, we assume that p is true and q is false and arrive at some sort of illogical statement such as 12. Intro to proofs proof by contrapositive involving an. Types of statements converse, inverse, contrapositive. A conditional statement consists of two parts, a hypothesis in the if clause and a conclusion in the then clause.
Before we define the converse, contrapositive, and inverse of a conditional statement, we need to examine the topic of negation. Converse, inverse, contrapositive given an ifthen statement if p, then q, we can create three related statements. Each implication implies its contrapositive, even intuitionistically. Contrapositive examples the infinite series module.
The basic concept is that proof by contrapositive relies on the fact that p. If 15 is not a prime number, then 15 is not an odd number. I am looking more for a set of guidelines or intuitions or heuristics that might suggest that trying to use the contrapositive to prove the mathematical statement might be a good idea. If the negation of a statement uses the word not, then disjunction uses the word or. Mar 23, 2016 this is a lesson excerpt from our online course, for which we invite you to enroll. It is very important to realize that these two implications are not logically equivalent. Shorser the contrapositive of the statement \a b i. Hence, contrapositive of a statement is described as reversal and negation both the statements. Were going to look at these three terms using this statement, if something is a dog, then it is a mammal.
Contrapositive proof example proposition suppose n 2z. The symbols for contrapositive may be read as not q implies not p or if not q, then not p. Converses, contrapositives and proof by the contrapositive the converse of the implication p q is the reverse implication q p. In fact, the contrapositive is true because the original statement is true. For instance, the proposition all cats are mammals can be restated as the conditional if something is a cat, then it is a mammal.
A contrapositive has truth value equivalent to the original statement. Jun 05, 2015 type of statement statements symbolism conditional if it is a whole number, then it is an integer. And an example of how even i screw up basic algebra at times. The proves the contrapositive of the original proposition. Nov 02, 2014 2 example questions using proof by contrapositive. By denition, the reverse of an implication means the same as the original implicationitself. Soda pdf merge tool allows you to combine pdf files in seconds. Formulas and equations also important info on slope learn with flashcards, games, and more for free. Now if you take a second to think about that, thats true.
Our pdf merger allows you to quickly combine multiple pdf files into one single pdf document, in just a few clicks. Combine pdfs in the order you want with the easiest pdf merger available. Whenever a conditional statement is true, its contrapositive is also true and vice versa. A proof is a valid argument that establishes the truth of a theorem as the conclusion statements in a proof can include the axioms. This webapp provides a simple way to merge pdf files. Soda pdf is the solution for users looking to merge multiple files into a single pdf document. Therefore, if you show that the contrapositive is true, you have also shown that the original statement is true. The statement \a implies b can be written symbolically. How to combine files into a pdf adobe acrobat dc tutorials. The contrapositive of the statement has its antecedent and consequent inverted and flipped. Since proof by contrapositive involves negating certain logical statements, one has to be careful. The method of proof by contraposition is based on the logical equivalence between a statement and its contrapositive. Combine multiple pdf files into one single pdf document.
A statement and its contrapositive are logically equivalent, in the sense that if the statement is true, then its contrapositive is true and vice versa. The underlying reasoning is that since a conditional statement is logically equivalent to its contrapositive, if the contrapositive is true, then the statement must also be true. Contraposition definition of contraposition by merriam. From the truth table, the columns of conditional statement and contrapositive statement are same. The contrapositive of a conditional statement always has the same truth value as the original statement. If ab 0 and bc sep 25, 2014 this video demonstrates a proof by contrapositive.
For instance, suppose we want to prove if mathamath, then mathbmath. Bc is a province belonging to canada, as shown in the diagram. If a product of two positive real numbers is greater than 100, then at least one of the number is greater than 10. This free online tool allows to combine multiple pdf or image files into a single pdf document. Maakt het mogelijk om pdfbestanden samen te voegen met een simpele draganddrop interface. Contraposition definition is the relationship between two propositions when the subject and predicate of one are respectively the negation of the predicate and the negation of the subject of the other. If youre having trouble with understanding what a contrapositive is on the lsat, then watch this short video and read the rest of the post. Find the truth values for the converse, inverse, and contrapositive of the implica tions stated below.
Hence, the given statement is true by the logical equivalence between a statement and its contrapositive. Type of statement statements symbolism conditional if it is a whole number, then it is an integer. Proof by contrapositive july 12, 2012 so far weve practiced some di erent techniques for writing proofs. To prove a statement of the form \if a, then b, do the following. The statement \a implies b can be written symbolically as \a b. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. You can either select the files you want to merge from you computer or drop them on the app using drag. The law of contrapositive says that a conditional statement is logically equivalent to its contrapositive. Pdf a statement, the contrapositive and the inverse. A free and open source software to merge, split, rotate and extract pages from pdf files.
1384 1480 103 852 1306 1621 1404 1041 1588 1429 865 780 150 1581 448 1521 1339 640 633 429 1064 954 257 294 110 514 958 1454 1509 238 429 1076 719 727 152 676 800 1266 627 1255 427 1124