Linear Methods in Discrete Mathematics [Lecture notes]

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It is a Contingency. Consider a sentence: x is greater than 2. Here is greater than 2 is the predicate and x is the subject or variable. If values are assigned to all the variables, the resulting sentence is a proposition. The set A is called domain of P x and the set Tp of all elements of A for which P a is true is called the truth set of P x.

Propositional functions can be converted to proposition by two aspects i By assigning exact value to the variable and ii using quantification. Quantifiers: Quantification is the way by which a Propositional function can be turns out to be a proposition. The process of applying quantifier to a variable is called quantification of variables. The symbol v is called the universal quantifier. This is true for any proposition p x. For example, The negation of all men are mortal is: There is a man who is not mortal. Then this implication is true regardless of the truth values of any of its components.

In this case, we say that Q logically follows from P1, P2…..

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Proofs in mathematics are valid arguments that establish the truth of mathematical statements. To deduce new statements from statements we already have, we use rules of i n f e r e n c e which are t e m p l a t e s for c o n s t r u c t i n g valid arguments. Rules of inference are our basic tools for establishing the truth of statements.

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  5. The rules of inference for statements involving existential and universal quantifiers play an important role in proofs in Computer Science and Mathematics, although they are often used without being explicitly mentioned. Valid Argument: An argument in propositional logic is a sequence of propositions. All propositions in the argument are called hypothesis or Premises. The final proposition is called the conclusion. An argument form in propositional logic is a sequence of compound propositions - involving propositional variables. An argument form is valid if no matter which particular propositions are substituted for the propositional variables in its premises, the conclusion is true if the premises are all true.

    Thus we say the conclusion C can be drawn from a given set of premises or the argument is valid if the conjunction of all the premises implies the conclusion is a tautology. Rules of Inference for Propositional logic We can always use a truth table to show that an argument form is valid. Their validity depends only on the form of statements involved and not on the truth values of the variables they contain such arguments are called rules of inference. These rules of inference can be used as building blocks to construct more complicated valid argument forms e.

    So, this is valid argument and hence is a rule of inference, called modus ponens or the law of detachment.

    Modus ponens is Latin for mode that affirms The most important rules of inference for propositional logic are as follows….. Example Test the validity of the following arguments : 1. If milk is black then every crow is white. If every crow is white then it has 4 legs. If every crow has 4 legs then every Buffalo is white and brisk. The milk is black.

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    So, every Buffalo is white. The following steps checks the validity of argument. Hypothetical syllogism H. P Premise iv 7. Either I will get good marks or I will not graduate. I get good marks. Thus, I would not go to USA. Solution : Let P : I will get good marks. Q : I will graduate. What are proofs? A proof is a clear explanation, accepted by the mathematical community, of why something is true. Ancient Babylonian and Egyptian mathematics had no proofs, just examples and methods. Let us discuss method of induction. The following result shows how this can be done.

    This result is called the principle of Mathematical induction. P n , using the principle of mathematical induction, we must begin by proving directly that the first proposition P n0 is true. This is called the basis step of the induction and is generally very easy. Since, the only case where an implication is false is if the antecedent This step is called induction step. In short we solve by following steps. Show that P 1 is true. Assume P k is true. Hence , by method of induction P n is true for all n. Let us also show that P 2 is true. Solution vi Statement P n is defined by n!

    STEP 2: We now assume that p k is true k!

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    COUNTING: Broadly speaking combinatory counting is the branch of mathematics dealing with order and patterns without regard to the intrinsic properties of the objects under consideration. For Example, suppose there are 3 routes from Burla to Sambalpur and 4 routes from Sambalpur to Cuttack, then by FPC the total number of ways for performing journey from Burla to Cuttack is Factorial Rule: For n items, there are n!

    Find the number of ways to text your friends. Solution: 4! There are 24 ways to test three out of your four friends if order matters. Example 3: The art club has 4 members. They want to choose a group of three to compete in a regional competition. How many ways can three members be chosen? Solution: There are 4 ways to chose 3 people for the competition when order is not important For Example, 13 people are involved in a survey to determine the month of their birthday. As we all know, there are 12 months in a year, thus, even if the first 12 people have their birthday from the month of January to the month of December, the 13th person has to have his birthday in any of the month of January to December as well.

    Thus, by PHP we are right to say that there are at least 2 people who have their birthday falling in the same month. In fact, we can view the problem as there are 12 pigeonholes months of the year with 13 pigeons the 13 persons. Of course, by the Pigeonhole Principle, there will be at least one pigeonhole with 2 or more pigeons. How many are wearing dresses that have only white color? Example: At Sunnydale High School there are 28 students in algebra class,30 students in biology class, and 8 students in both classes.

    How many students are in either algebra or biology class? Solution: Let A denote the set of students in algebra class and B denote the set of students in biology class. Example: At Sunnydale High School there are 55 students in either algebra, biology, or chemistry class 28 students in algebra class, 30 students in biology class, 24 students in chemistry class, 8 students in both algebra and biology, 16 students in both biology and chemistry, 5 students in both algebra and chemistry.

    How many students are in all three classes? Solution: Let A, B, C denote the set of students in algebra, biology, and chemistry class, Respectively. To count the number of Students in all three classes, i.

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    But for students in all three classes, we counted them 3 times, then subtracted them 3 times. But there are some problems which cannot be solved or very tedious to solve, using these techniques.

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    In some such problems, the problems can be represented in the form of some relation and can be solved accordingly. We shall discuss some such examples before proceeding further. The expression of higher terms in terms of combination of lower terms is known as recurrence relation Example: The number of bacteria, double every hour, then what will be the population of the bacteria after 10 hours? Here we can represent number of bacteria at the nth hour be an.

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    Example : Towers of Hanoi is a popular puzzle. There are three pegs mounted on a board, together with disks of different sizes. Initially, these discs are placed on the first peg in order of different sizes, with the largest disc at the bottom and the smallest at the top.

    The task is to move the discs from the first peg to the third peg using the middle peg as auxiliary. This is a popular puzzle and we shall discuss its solution, using the one of the techniques discussed in this chapter. With these illustrations, we define recurrence relation now. The term a0, given in the above two examples, specify initial condition to solve the recurrence relation completely. The first example of formulation that we discuss is the problem of Tower of Hanoi as above. Example: With reference to above Example, let Hn denote the number of moves required to solve the puzzle with n discs.

    Let us define Hn recursively.

    Linear Methods in Discrete Mathematics [Lecture notes] Linear Methods in Discrete Mathematics [Lecture notes]
    Linear Methods in Discrete Mathematics [Lecture notes] Linear Methods in Discrete Mathematics [Lecture notes]
    Linear Methods in Discrete Mathematics [Lecture notes] Linear Methods in Discrete Mathematics [Lecture notes]
    Linear Methods in Discrete Mathematics [Lecture notes] Linear Methods in Discrete Mathematics [Lecture notes]
    Linear Methods in Discrete Mathematics [Lecture notes] Linear Methods in Discrete Mathematics [Lecture notes]
    Linear Methods in Discrete Mathematics [Lecture notes] Linear Methods in Discrete Mathematics [Lecture notes]
    Linear Methods in Discrete Mathematics [Lecture notes] Linear Methods in Discrete Mathematics [Lecture notes]

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