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creativity in every individual with a different set of talents through the basic understanding of humanity and nature. Mathematics is the subject dedicated to. General Mathematics - books for free online reading: introductory works and textbooks. pp, MB, PDF. Basic Concepts of Mathematics by Elias Zakon , , Of Mathematics by Viatcheslav Vinogradov, , pages, KB, PDF. COURSE. Mathematics, Basic Math and. Algebra. NAVEDTRA . III Mathematical symbols. .. and p are more general than the numbers 9, 8, and 7.
Mathematics for Materials Scientists and Engineers by W. Write the following propositions symbolically: Writing Style: If one considers sets such as pigs, cows, chickens, or horses, the universal set is probably the set of animals. We are given the following information: Case 1:
Determine a. For convenience, we use A to denote the complement of a set A. Find all the subsets of S. Let A and B be sets. Let A, B and C be sets. Prove by the Elements Argument method that a. Find a. Let A and B be non-empty sets. Draw Venn diagrams for each of the following sets. Shade the region corresponding to each set.
In a survey of households, owned a home computer, a video, two cars, and households owned neither a home computer nor a video nor two cars.
In a survey conducted on campus, it was found that students like watching the Barclays Premier League teams: ManU, Chelsea and Arsenal. It was also found that every student who is a fan of Arsenal is also a fan of ManU or Chlesea or both , and 42 students were fans of ManU, 45 were fans of Chelsea, 7 where fans of both ManU and Chelsea, 11 were fans of of both ManU and Arsenal, 28 were fans of both Chelsea and Arsenal, and twice as many students were fans only of ManU as those who were fans only of Chelsea.
Find the number of students in the survey who were fans of a. Arsenal c. Find the following power sets a. The need for complex numbers must have been felt from the time that the formula for solving quadratic equations was discovered, especially due to the existence of square roots of negative numbers. The set of real numbers might seem to be a large enough set of numbers to answer all our mathematical questions adequately. However, there are some natural mathematical questions that have no solu- tion if answers are restricted to be real numbers.
In particular, many simple equations have no solution in the realm of real numbers. A solution would require a number whose square is 1. However, di- vision of complex numbers is not straightforward.
We need to develop the theory to enable us carry out division of complex numbers. Two complex numbers are equal if and only if they have the same real part and the same imaginary part. Note that 0 is the only number which is at once real and purely imaginary. A similar relation exists between the set of points in the plane and the set of complex numbers.
When a plane is used in this way to picture complex numbers, it is called the complex number plane. It is also called the Argand Diagram after J. The horizontal axis of the Argand Diagram is called the real axis and the vertical axis is called the imaginary axis.
This process is called complex rationalization. Find z. Cube roots of unity Example 1. Mark on the Argand diagram the points representing the complex numbers a.
The subject has origins in philosophy, and indeed it is also a legacy from philosophy that we can distinguish semantic reasoning what is true from syntactic reasoning what can be shown.
Logic is used to establish the validity of arguments. The rules of logic give pre- cise meaning to mathematical statements.
For example, a young child touches a hot stove and concludes that stoves are hot. It is usually by inductive reasoning that mathematical results are discovered, and it is by deductive reasoning that they are proved.
Inductive Reasoning Inductive reasoning is essential to mathematical activity. To engage in it, one makes observations, gets hunches, guesses, or makes conjectures. Example 2. Solution It is probably 10, etc. Deductive Reasoning Arguments used in mathematical proofs most often proceed from some basic principles which are known or assumed.
Such arguments are deductive. Figure X is a triangle. What conclusion can be drawn? This system is built around propositions or statements.
Propositions are sometimes called statements. It is raining 2. Nairobi is the capital city of Rwanda 4. Tomorrow is my birthday Remark 2. Whilst proposition 5 is true when stated by anyone whose birthday is tomorrow is true, it is false when stated by anyone else.
Come here! Long live the Queen! The truth value of a proposition is either true or false but not both. We denote the truth vales of propositions by T or F. Propositions are conventionally symbolized using letters a, b, c, In this subsection we look at how simple propositions can be combined to form more complicated propositions called compound statements.
The devices we use are link pairs or more propositions are called logical connectives. For example, let P denote the proposition: Then the following are some of its negations: In accordance with ordinary language, the negation of a true proposition will be consid- ered false, and the negation of a false proposition will be considered a true proposition. The truth table of a negation is given by We consider four commonly used logical connectives: Truth Table of a Conjunction 2.
The compound proposition so formed is called a disjunction of P and Q. P and Q are called dijuncts. This compound statement is true when either or both of its components are true and is false otherwise. The truth table of a disjunction is give by Figure 2. This compound proposition is true when exactly one i.
The truth table of of P Y Q is given by Figure 2. Truth Table of an Exclusive Disjunction The context of a disjunction will often provide the clue as to whether the inclusive or exclusive sense is intended. Such propositions are extremely important in mathematical proofs. In a deduc- tive argument, something is assumed and something is concluded. Let P and Q are propositions. Truth Table of an Implication Example 2. I eat breakfast Q: Let P and Q be propositions. Mathematician are generous.
Spiders hate algebra.
Write the compound propositions symbolized by a. It is not the case that spiders hate algebra and mathematicians are generous. If Mathematicians are not generous then spiders hate algebra. Write the following propositions symbolically: Today is Monday or I will go to London but not both c. I will go to London and today is not Monday. If and only if today is not Monday then I will go to London.
Left as an exercise. A contradiction is a compound proposition which is false no matter what the truth values of its simple components are. We shall denote a tautology by t and a contradiction by c. Their truth tables return a column true values T.
Their truth tables return a column false values F. Solution The truth table for the two propositions is given below. Figure 2. Logical Equivalence The last two columns are the same and hence the two propositions are logically equiva- lent.
Remark 2. Truth Table for a Conditional and its Converse, Inverse and Contrapositive From the table, we note the following useful results: Jack plays his guitar. If Jack plays his guitar then Sarah will sing. If Sara will sing then Jack plays his guitar. An open sentence is also called a predicate.
A predicate is a statement p x1 , x2 , We consider the truth values of p 1 , p 2 , We need not check any other values from U. That is, there is at least one rat which is is grey. That is, there is at least one x that does not have the property p.
All men are mortal. Some men live in the city. Many mathe- matical theorem are statements that a certain implication is true. We give some methods of proof. This is called proof by a counter-example. Sometimes, it is possible to prove such a statement directly; that is, by establishing the validity of a sequence of implications: Proof We can consider the cases: Case 1: The product of an even integer and any integer is even.
Since n is even, 9n2 and 3n are even too. Case 2: The product of odd integers is odd. If we can establish the truth of the contrapositive, we can deduce that the conditional is also true.
Proof Let P and Q be the statements P: Call given integers a, b, c, d. So the biggest possible average would be 38 4 , which is less than 10, so P is false. We simply do not know how to begin. In this case, we sometimes make progress by assuming that the negation of P is true. So P must be true.
If P is false, then there is a largest integer N. It consists of the following steps: Theorem 2. In many applications, m will be 0 or 1. Condition i is called the basis, and ii is called the inductive step.
The basis is easy to check; the inductive step is sometimes quite a bit more complicated to verify. The principle tells that if we can show that i and ii holds, we are done. Assume inductively that p k is true for some positive integer k. In each of the following, construct the conjunction and disjunction of the set of simple propositions.
Decide if you can, the truth value of each compound statement. July has 29 days. Christmas is December 25th. Let P, Q and R be propositions. Construct a truth table for a. Write the negation of each of the following propositions. Determine whether each of the following is a tautology, a contradiction or neither.
Simplify each of the following propositions, quoting the laws you use.
Which of the following are propositions a. As the world turns. An apple a day keeps the doctor away c. Are my parents obligated to buy me a car? Give a reason for your answer. Suppose you order a chicken sandwich at a Kenchic restaurant. The waitress tells you that the sandwich comes with soup or salad. Is the waitress most likely to be using an inclusive OR or an exclusive OR?
Consider the propositions p: Felix laughs q: Jacinta cries r: John shouts Write in words the following compound propositions a. Bats are blind q: Sheep eat grass r: Ants have long teeth Express the following compound propositions symbolically a.
Give a direct proof that if n is odd then n2 is odd. Find a counterexample to the proposition: They are types of arrangements of elements of a set. Counting el- ements in a probability problem or occurrence problem individually may be extremely tedious or even prohibitive.
We begin by motivating the basic counting principle which is useful in solving a wide variety of problems. Theorem 3. Solution Figure 3. Routes from A to D Here we have a three-stage procedure: By the Basic Counting Principle, the total number of routes is 2.
Example 3. Solution The procedure of labeling a chair consists of two tasks, namely, assigning one of the letters and the assigning one of the possible integers.
The quiz consists of three multiple-choice questions with four choices for each. Successively answering the three questions is a three-stage procedure. Likewise, each of the other two questions can be answered in four ways. By the Basic Counting Principle, the number of ways to answer the quiz is 4.
Answering the quiz can be considered a two-stage procedure. From part a , the three multiple-choice questions can be answered in 4.
Each of the true-false questions has two choices true or false. By the BCP, the number of ways the entire quiz can be answered is This is a three-stage procedure.
By the Basic Counting Principle, the total number of three-letter words is 5. For instance Example 3. Thus we have the following result: The number of permutations of n objects taken r at a time is given by n! Solution We shall consider a slate in the order of president, vice president, secretary and treasurer. Each ordering of four members constitutes a slate, so the number of possible slates is 20!
The other nine can then be arranged in 9! Figure 3. Arrangement on a round table 3. Thus, the number of distinguishable permutations is not 7! We now give a formula to enable us solve similar problems. Using this formula, the answer to Example 3.
Note that the word 7! Therefore there are 3! Massasauga is a white venomous snake indigenous in North America. Note that this word has 10 characters, some repeated: Therefore there are 10! Now we have 7 characters with some characters repeated: Therefore there are 7!
Remark 3. In how many ways can this be done? Obviously, order in which people are placed into the rooms is of no concern. The cells remind us of those permutations with repeated objects. In how many ways can this be done. Solution Here people are placed into three cells matatus: Thus there are 15! Solution The combinations are: The permutations are: Listing all such combinations and all permutations of these combinations, we obtain a list of permutations of the n objects taken r at a time.
Thus n n! Solution Order is not important because no matter how the members of a committee are ar- ranged, we have the same committee. Thus, we simply have to compute the number of combinations of 20 objects taken four at a time, 20 20! Solution Once the group of 8 has been selected then the remaining 12 children will automatically comprise the other group. For the selection of those to join the group of 8 we have two cases: The total number of ways will be 18 18 18!
There are bananas, apples, pears, kwi, apricots, and oranges in the house. In how many ways can a selection of four pieces of fruit be chosen? Solution Note that only the selection of varieties not which person eats what fruit is of interest here.
This is a combination with repetition problem. Thus the solution is 9! How many distinguishable arrangements of the word are possible? The number of distinguishable arrangements is 11! We must choose three men from 20 and two women from This can be done in 20 The answer is 12 20 12 Use your result to approximate 1.
Note that 1. Therefore 1.
Solution Using Theorem 3. Surowski, , pp, 3. Basic Concepts of Mathematics by Elias Zakon, , pages, 1. Blast Into Math! Business Mathematics: A Textbook by Edward I. Edgerton, Wallace E. Bartholomew, , pp, multiple formats. Encyclopedia of Mathematics Kluwer Academic Publishers, , online html. Engineering Mathematics with Tables by M. Keasey, G. Kline, D. McIlhatten, Engineering Mathematics: YouTube Workbook by Christopher C. Tisdell, , pp, 2. Essential Engineering Mathematics by Michael Batty, , pages, 4.
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