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An introduction to the fundamental concepts of calculus, including the notion of limits. It covers examples of calculating limits for various functions, and introduces the concepts of horizontal and vertical asymptotes. The document also discusses the properties of logarithms and their applications in solving equations and inequalities. Additionally, it covers the basic trigonometric functions and their inverse functions, as well as the properties of periodic functions. Overall, this document serves as a comprehensive introduction to the key topics in calculus, equipping students with the necessary knowledge and problem-solving skills to progress in their studies.
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MOSHE
S E P T E M B E R , 2 0 1 9
Chapter One
Propositional Logic and Set Theory
In this chapter, we study the basic concepts of propositional logic and some part of set theory. In the first part, we deal about propositional logic, logical connectives, quantifiers and arguments. In the second part, we turn our attention to set theory and discus about description of sets and operations of sets.
Know methods and procedures in combining the validity of statements. Understand the concept of quantifiers. Know basic facts about argument and validity. Understand the concept of set. Apply rules of operations on sets to find the result. Show set operations using Venn diagrams.
1.1. Propositional Logic Mathematical or symbolic logic is an analytical theory of the art of reasoning whose goal is to
systematize and codify principles of valid reasoning. It has emerged from a study of the use of language
in argument and persuasion and is based on the identification and examination of those parts of
language which are essential for these purposes. It is formal in the sense that it lacks reference to
meaning. Thereby it achieves versatility: it may be used to judge the correctness of a chain of reasoning
(in particular, a "mathematical proof") solely on the basis of the form (and not the content) of the
sequence of statements which make up the chain. There is a variety of symbolic logics. We shall be
concerned only with that one which encompasses most of the deductions of the sort encountered in
mathematics. Within the context of logic itself, this is "classical" symbolic logic.
Section objectives:
After completing this section, students will be able to:-
Identify the difference between proposition and sentence. Describe the five logical connectives.
Determine the truth values of propositions using the rules of logical connectives. Construct compound propositions using the five logical connectives. Determine the truth values of compound propositions. Distinguish a given compound proposition is whether tautology or contradiction.
1.1.1. Definition and examples of propositions Consider the following sentences.
The first three sentences are declarative sentences. The first one is true and the second one is false. The truth value of the third sentence cannot be ascertained because of lack of historical records but it is, by its very form, either true or false but not both. On the other hand, the last three sentences have not truth value. So they are not declaratives.
Now we begin by examining proposition, the building blocks of every argument. A proposition is a sentence that may be asserted or denied. Proposition in this way are different from questions, commands, and exclamations. Neither questions, which can be asked, nor exclamations, which can be uttered, can possibly be asserted or denied. Only propositions assert that something is (or is not) the case, and therefore only they can be true or false.
Definition 1.1: A proposition (or statement) is a sentence which has a truth value (either True or False but not both).
The above definition does not mean that we must always know what the truth value is. For example, the sentence “The 1000th^ digit in the decimal expansion of is 7” is a proposition, but it may be necessary to find this information in a Web site on the Internet to determine whether this statement is true. Indeed, for a sentence to be a proposition (or a statement), it is not a requirement that we be able to determine its truth value.
Remark: Every proposition has a truth value, namely true (denoted by ) or false (denoted by ).
1.1.2. Logical connectives
In mathematical discourse and elsewhere one constantly encounters declarative sentences which have been formed by modifying a sentence with the word “not” or by connecting sentences with the words “and”, “or”, “if... then (or implies)”, and “if and only if”. These five words or combinations of words are called propositional connectives.
Example 1.2: Consider the following propositions:
: 3 is an odd number. (True)
: 27 is a prime number. (False)
: Nairobi is the capital city of Ethiopia. (False)
Note: The use of “ or ” in propositional logic is rather different from its normal use in the English language. For example, if Solomon says, “I will go to the football match in the afternoon or I will go to the cinema in the afternoon,” he means he will do one thing or the other, but not both. Here “or” is used in the exclusive sense. But in propositional logic, “or” is used in the inclusive sense; that is, we allow Solomon the possibility of doing both things without him being inconsistent.
Implication
When two propositions are joined with the connective “ implies ,” the proposition formed is called a logical implication. “implies” is denoted by “ .” So, the logical implication of two propositions, and , is written:
read as “ implies .”
The function of the connective “implies” between two propositions is the same as the use of “If … then …” Thus can be read as “if , then .”
is false if and only if is true and is false.
This form of a proposition is common in mathematics. The proposition is called the hypothesis or the antecedent of the conditional proposition while is called its conclusion or the consequent.
The following is the truth table for implication.
Examples 1.3: Consider the following propositions:
: 3 is an odd number. (True)
: 27 is a prime number. (False)
: Addis Ababa is the capital city of Ethiopia. (True)
: If 3 is an odd number, then 27 is prime. (False)
: If 3 is an odd number, then Addis Ababa is the capital city of Ethiopia. (True)
We have already mentioned that the implication can be expressed as both “If , then ” and “ implies .” There are various ways of expressing the proposition , namely:
If , then.
if.
implies.
only if.
is sufficient for.
is necessary for
Bi-implication
When two propositions are joined with the connective “ bi-implication ,” the proposition formed is called a logical bi-implication or a logical equivalence. A bi-implication is denoted by “ ”. So the logical bi- implication of two propositions, and , is written:
.
is false if and only if and have different truth values.
Example 1.5: Let : Addis Ababa is the capital city of Ethiopia. (True)
: Addis Ababa is not the capital city of Ethiopia. (False)
1.1.3. Compound (or complex) propositions
So far, what we have done is simply to define the logical connectives, and express them through algebraic symbols. Now we shall learn how to form propositions involving more than one connective, and how to determine the truth values of such propositions.
Definition 1.2: The proposition formed by joining two or more proposition by connective(s) is called a compound statement.
Note: We must be careful to insert the brackets in proper places, just as we do in arithmetic. For example, the expression will be meaningless unless we know which connective should apply first. It could mean or , which are very different propositions. The truth value of such complicated propositions is determined by systematic applications of the rules for the connectives.
The possible truth values of a proposition are often listed in a table, called a truth table. If and are propositions, then there are four possible combinations of truth values for and. That is, , , and. If a third proposition is involved, then there are eight possible combinations of truth values for , and. In general, a truth table involving “ ” propositions , ,…, contains possible combinations of truth values for these propositions and a truth table showing these combinations would have columns and rows. So, we use truth tables to determine the truth value of a compound proposition based on the truth value of its constituent component propositions.
Examples 1.6:
What is the truth value of?
and that the truth values of and are and , respectively. Then the truth value of is , that of is , that of is. So the truth value of is.
Remark: When dealing with compound propositions, we shall adopt the following convention on the use of parenthesis. Whenever “ ” or “ ” occur with “ ” or “ ”, we shall assume that “ ” or “ ” is applied first, and then “ ” or “ ” is then applied. For example,
means
It is useful at this point to mention the non-equivalence of certain conditional propositions. Given the conditional , we give the related conditional propositions:-
: Converse of
: Inverse of
: Contrapositive of
As we observed from example 1.7, the conditional and its contrapositve are equivalent. On the other hand, and.
Do not confuse the contrapositive and the converse of the conditional proposition. Here is the difference:
Converse: The hypothesis of a converse statement is the conclusion of the conditional statement and the conclusion of the converse statement is the hypothesis of the conditional statement.
Contrapositive: The hypothesis of a contrapositive statement is the negation of conclusion of the conditional statement and the conclusion of the contrapositive statement is the negation of hypothesis of the conditional statement.
Example 1.9:
Propositions, under the relation of logical equivalence, satisfy various laws or identities, which are listed below.
1.1.4. Tautology and contradiction
Definition: A compound proposition is a tautology if it is always true regardless of the truth values of its component propositions. If, on the other hand, a compound proposition is always false regardless of its component propositions, we say that such a proposition is a contradiction.
Examples 1.10:
1.2. Open propositions and quantifiers
In mathematics, one frequently comes across sentences that involve a variable. For example, is one such. The truth value of this statement depends on the value we assign for the
variable. For example, if , then this sentence is true, whereas if , then the sentence is false.
Section objectives:
After completing this section, students will be able to:-
Define open proposition. Analyze the difference between proposition and open proposition. Differentiate the two types of quantifiers. Convert open propositions into propositions using quantifiers. Determine the truth value of a quantified proposition. Convert a quantified proposition into words and vise versa. Explain the relationship between existential and universal quantifiers. Analyze quantifiers occurring in combinations.
Definition 1.4: An open statement (also called a predicate) is a sentence that contains one or more variables and whose truth value depends on the values assigned for the variables. We represent an open statement by a capital letter followed by the variable(s) in parenthesis, e.g., etc.
Example 1.11: Here are some open propositions:
It is clear that each one of these examples involves variables, but is not a proposition as we cannot assign a truth value to it. However, if individuals are substituted for the variables, then each one of them is a proposition or statement. For example, we may have the following.
Remark
The collection of all allowable values for the variable in an open sentence is called the universal set (the universe of discourse) and denoted by.
Now we proceed to study open propositions which are satisfied by “ all ” and “ some ” members of the given universe.
and "for each " as having the same meaning and symbolize each by “ .” Think of the symbol as an inverted (representing all). If is an open proposition with universe , then is a quantified proposition and is read as “every has the property .”
"for some ," and "for at least one " as having the same meaning, and symbolize each by “ .” Think of the symbol as the backwards capital (representing exists). If is an open proposition with universe , then is a quantified proposition and is read as “there exists with the property .”
Remarks :
ii. is if we cannot find any having the property.
Example 1.14:
Relationship between the existential and universal quantifiers
If is a formula in , consider the following four statements.
We might translate these into words as follows.
Now (d) is the denial of (a), and (c) is the denial of (b), on the basis of everyday meaning. Thus, for example, the existential quantifier may be defined in terms of the universal quantifier.
Now we proceed to discuss the negation of quantifiers. Let be an open proposition. Then is false only if we can find an individual “ ” in the universe such that is false. If we succeed in getting such an individual, then is true. Hence will be false if is true. Therefore the negation of is. Hence we conclude that
.
Similarly, we can easily verified that
.
Remark: To negate a statement that involves the quantifiers and , change each to , change each to , and negate the open statement.
Example 1.15:
Let.
Given propositions containing quantifiers we can form a compound proposition by joining them with connectives in the same way we form a compound proposition without quantifiers. For example, if we have and we can form.
Consider the following statements involving quantifiers. Illustrations of these along with translations appear below.
