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How many presidents had at most one degree? That is,exactly of the presidents had both undergraduate and postgraduate degrees in business. That is, exactly of the presidents had no undergraduate degree and no postgraduate degree in business. The problem can be represented in A Venn diagram as below: At most one degree means either undergraduate or postgraduate degrees but not both.
Draw a Venn diagram to represent this information Using your Venn diagram together with the Inclusion-Exclusion Principle where need be , determine b. At least two of the media means we add: Exactly one of the media means Radio only or Television only or Newspapers only. From the Venn diagram we have: We are given the following information: Now consider set M.
However, this is not necessary to answer the three parts of the question.
We begin this section by studying the decomposition of the real line into the following subsets: Note that N is closed with respect to the usual addition and usual multiplication but not with respect to usual subtraction.
The set of natural numbers consists of the dark points on Fig. Set of Natural numbers 1. Note that W is closed with respect to usual addition and multiplication but not under subtraction. The set of whole numbers consist of the numbers represented by the dark dots in Fig 1. Set of Whole numbers 1. Z is closed with respect to usual addition and usual multiplication. Set of Integers 1. Note that the sets of rationals and irrationals are disjoint. Graphically, R is represented by the real number line and called the real number system.
This means that a rational number is either rational or irrational but not both. Set of Real numbers Every point on the real line represents a real number. 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.