Discrete mathematics pdf notes dm lecture notes pdf. No knowledge about monopoly was required to determine that the statement was true. Tautology in math definition, logic, truth table and examples. Nov 15, 2017 tautology contradiction contingency satisfiability propositional logic gate net part 6. Example sheets 1 and 2, covering propositional and predicate logic. Take this interactive quiz and test your understanding of a tautology. Discrete mathematics propositional logic in discrete. Thus, completely devoid is an example of tautology.
A proposition is said to be a contradiction if its truth value is f for any assignment of truth values to its components. This book is designed for a one semester course in discrete mathematics for sophomore or junior level students. A statement whose form is a tautology is a tautological statement. Jul 25, 2019 tautology, contradiction and contingency. Lecture notes on discrete mathematics july 30, 2019. Discrete mathematics and its applications lecture 1. A tautology is an expression or phrase that says the same thing twice, just in a different way. Truthtables,tautologies,andlogicalequivalences mathematicians normally use a twovalued logic. In general one can check whether a given propositional formula is a tautology by simply examining its truth table. The text covers the mathematical concepts that students will encounter in many disciplines such as computer science, engineering, business, and the sciences. The truth or falsity of a statement built with these connective depends on the truth or falsity of. Propositional logic studies the ways statements can interact with each other.
In my last video we have seen converse, inverse and contrapositive of an implication and its examples. Discrete mathematics discrete mathematics study of mathematical structures and objects that are fundamentally discrete rather than continuous. Tautologies are always true but they dont tell us much about the world. The discrete mathematics notes pdf dm notes pdf book starts with the topics covering logic and proof, strong induction,pigeon hole principle, isolated vertex, directed graph, alebric structers. The compound statement p p consists of the individual statements p and p. Repeating an idea in a different way can bring attention to the idea. A statement in sentential logic is built from simple statements using the logical connectives,, and. Richard mayr university of edinburgh, uk discrete mathematics. The opposite of tautology is contradiction or fallacy which we will learn here. This tautology, called the law of excluded middle, is a. Say p is a tautology, or is valid, if it is always true i. Discrete mathematics about the tutorial discrete mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. Intuitively, if we have the condition of an implication, then we can obtain its consequence.
For example, in terms of propositional logic, the claims, if the moon is made of cheese then basketballs are round, and if spiders have eight legs then sam walks. This tautology, called the law of excluded middle, is a direct consequence of our basic assumption that a proposition is a statement that is either true or false. Some authorities say repetition uses the same words, while tautology uses words with similar meanings. Similarly, if you have a compound statement, s, of the form f. However, there are times when tautology is done for effect. A tautology in math and logic is a compound statement premise and conclusion that always produces truth. If someone learns discrete mathematics, then they will find a good job. Propositions p and q are logically equivalent if p q is a tautology. A compound propositioncan be created from other propositions using logical connectives.
Others say there is no clear distinction between the two, that tautology includes the repetition. Discrete structures lecture notes vladlen koltun1 winter 2008 1computer science department, 353 serra mall, gates 374, stanford university, stanford, ca 94305, usa. Let us look at the classic example of a tautology, p. Others say there is no clear distinction between the two, that tautology includes the repetition of words. For this reason, a tautology is usually undesirable, as it can make you sound wordier than you need to be, and make you appear foolish. Tautology contradiction contingency satisfiability. This enforces that the truth value of p and the truth value of q must always be the same.
For example, if we have a finite set of objects, the function can be defined as a list of ordered pairs having these objects, and can be presented as a complete list of those pairs. A compound proposition that is always false is called a contradiction. A tautology is a compound statement in maths which always results in truth value. Tautology is when something is repeated, but it is said using different words. Browse other questions tagged discrete mathematics logic or ask your own question. It is important to remember that propositional logic does not really care about the content of the statements. A tautology is a formula which is always true for every value of its propositional variables. Discrete mathematics propositional logic in discrete mathematics discrete mathematics propositional logic in discrete mathematics courses with reference manuals and examples pdf. The word tautology is derived from a greek word where tauto means same and logy means logic. In propositional logic, we have a connective that combines two propositions into a new proposition called the conditional, or implication of the originals, that attempts to capture the sense of such a statement. If you not still watched that video, please watch that video before watching this video. A compound statement is made with two more simple statements by using some conditional words such as and, or, not, if, then, and if and only if. If i will study databases, then i will study computer science.
The statement about monopoly is an example of a tautology, a statement which is true on the basis of its logical form alone. A proposition that is neither a tautology nor a contradiction is. This is called the law of the excluded middle a statement in sentential logic is built from simple statements using the logical connectives,, and. It doesnt matter what the individual part consists of, the result in tautology is always true. A tautology is a compound statement which is true for every value of the individual statements. A tautology can reveal important information about an assertion. Introduction to logic introduction i introduction ii examples i. Discrete mathematics lecture notes, yale university, spring 1999 l. There are times when repetition is accidentalthe writer or speaker did not mean to repeat the idea. Predicate logic and quanti ers computer science and. It is increasingly being applied in the practical fields of mathematics and computer science. Its true that whether every mathematical theorem is a tautology depends on the notion of tautology being used.
Therefore, we conclude that p p is a tautology definition. Some of the examples were left as exercise for you. If you like geeksforgeeks and would like to contribute, you can also write an article using contribute. Vesztergombi parts of these lecture notes are based on l. In other words, a contradiction is false for every assignment of truth values to its simple components. Here you can download the free lecture notes of discrete mathematics pdf notes dm notes pdf materials with multiple file links to download. In the recursive definition of a set, the first rule is the basis of recursion, the second rule. Let pbe the statement maria learns discrete mathematics. The opposite of a tautology is a contradiction or a fallacy, which is always false. Discrete mathematics propositional logic the rules of mathematical logic specify methods of reasoning mathematical statements. The most useful class of tautologies are logical equivalences. A proposition that is neither a tautology nor a contradiction is called a contingency.
Existence proof examples show that there is a positive integer that can be written as the sum of cubes of positive integers in two different ways. The opposite of a tautology is a contradiction, a formula which is always false. Discrete mathematics and its applications, by kenneth h rosen this article is contributed by chirag manwani. Browse other questions tagged discretemathematics logic or ask your own. Discrete mathematics propositional logic tutorialspoint. Examples of objectswith discrete values are integers, graphs, or statements in logic. Wuct121 discrete mathematics logic tutorial exercises solutions. A compound statement, that is always true regardless of the truth value of the individual statements, is defined to be a tautology.
To understand this better, read the following examples of tautology. There is, however, a consistent logical system, known as constructivist. Propositional logic basics propositional equivalences normal forms boolean functions and digital circuits propositional equivalences. If you construct a truth table for a statement and all of the column values for the statement are true t, then the statement is a. Greek philosopher, aristotle, was the pioneer of logical reasoning. In the truth table above, p p is always true, regardless of the truth value of the individual statements.
Examples of tautology a tautology is an expression or phrase that says the same thing twice, just in a different way. This is a course on discrete mathematics as used in computer. If i will study discrete math, then i will study computer science. Predicate logic and quanti ers cse235 predicate logic and quanti ers slides by christopher m. Mathematics propositional equivalences geeksforgeeks. Mathematics is the only instructional material that can be presented in an entirely undogmatic way. It deals with continuous functions, differential and integral calculus. Two propositions p and q are logically equivalent if p q is a tautology.
It is usual to give a presentation of propositional calculus which is both sound. Occasionally, a tautology can help to add emphasis or clarity, or introduce intentional ambiguity, but in most cases its best to choose just one way to state your. Besides reading the book, students are strongly encouraged to do all the. Wuct121 discrete mathematics logic tutorial exercises.