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Chapter 1
Logic and Sets
1.1. Logical connectives
1.1.1. Unambiguous statements. Logic is concerned first of all with the logical structure of statements, and with the construction of complex statements from simple parts. A statement is a declarative sentence, which is supposed to be either true or false. A statement must be made completely unambiguous in order to be judged as true or false. Often this requires that the writer of a sentence has es- tablished an adequate context which allows the reader to identify all those things referred to in the sentence. For example, if you read in a narrative: “He is John’s brother,” you will not be able to understand this simple as- sertion unless the author has already identified John, and also allowed you to know who “ he ” is supposed to be. Likewise, if someone gives you direc- tions, starting “Turn left at the corner,” you will be quite confused unless the speaker also tells you what corner and from what direction you are sup- posed to approach this corner. The same thing happens in mathematical writing. If you run across the sentence x^2 ≥ 0 , you won’t know what to make of it, unless the author has established what x is supposed to be. If the author has written, “Let x be any real number. Then x^2 ≥ 0 ,” then you can understand the statement, and see that it is true. A sentence containing variables, which is capable of becoming an an unambiguous statement when the variables have been adequately identi- fied, is called a predicate or, perhaps less pretentiously, a statement-with- variables. Such a sentence is neither true nor false (nor comprehensible) until the variables have been identified. It is the job of every writer of mathematics (you, for example!) to strive to abolish ambiguity. The first rule of mathematical writing is this: any
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2 1. LOGIC AND SETS
symbol you use, and any object of any sort to which you refer, must be ad- equately identified. Otherwise, what you write will be meaningless or in- comprehensible.
Our first task will be to examine how simple statements can be combined or modified by means of logical connectives to form new statements; the validity of such a composite statement depends only on the validity of its simple components. The basic logical connectives are and , or , not , and if...then. We con- sider these in turn.
1.1.2. The conjunction and****. For statements A and B, the statement “A and B” is true exactly when both A and B are true. This is convention- ally illustrated by a truth table :
A B A and B t t t t f f f t f f f f
The table contains one row for each of the four possible combinations of truth values of A and B; the last entry of each row is the truth value of “A and B” corresponding to the given truth values of A and B. For example:
- “Julius Caesar was the first Roman emperor, and Wilhelm II was the last German emperor” is true, because both parts are true.
- “Julius Caesar was the first Roman emperor, and Peter the Great was the last German emperor” is false because the second part is false.
- “Julius Caesar was the first Roman emperor, and the Seventeenth of May is Norwegian independence day.” is true, because both parts are true, but it is a fairly ridiculous statement.
- “ 2 < 3 , and π is the area of a circle of radius 1” is true because both parts are true.
1.1.3. The disjunction or****. For statements A and B, the statement “A or B” is true when at least one of the component statements is true. Here is the truth table:
4 1. LOGIC AND SETS
A B A and B not(A and B) t t t f t f f t f t f t f f f t
Next, we observe that “not(A) or not(B)” has the same truth table as “not(A and B).”
A B not(A) not(B) not(A) or not(B) t t f f f t f f t t f t t f t f f t t t
We say that two statement formulas such as “not(A and B)” and “not(A) or not(B)” are logically equivalent if they have the same truth table; when we substitute actual statements for A and B in the logically equivalent state- ment formulas, we end up with two composite statements with exactly the same truth value; that is one is true if, and only if, the other is true. What we have verified with truth tables also makes perfect intuitive sense: “A and B” is false precisely if not both A and B are true, that is when one or the other, or both, of A and B is false.
Exercise 1.1.1. Check similarly that “not(A or B)” is logically equiv- alent to “not(A) and not(B),” by writing out truth tables. Also verify that “not(not(A))” is equivalent to “A,” by using truth tables.
The logical equivalence of “not(A or B)” and ‘not(A) and not(B)” also makes intuitive sense. “A or B” is true when at least one of A and B is true. “A or B” is false when neither A nor B is true, that is when both are false. Examples:
- The negation of “Julius Caesar was the first Roman emperor, and Wilhelm II was the last German emperor” is “Julius Caesar was not the first Roman emperor, or Wilhelm II was not the last German emperor.” This is false.
- The negation of “Julius Caesar was the first Roman emperor, and Peter the Great was the last German emperor” is “Julius Caesar was not the first Roman emperor, or Peter the Great was not the last German emperor.” This is true.
- The negation of “Julius Caesar was the first Chinese emperor, or Peter the Great was the last German emperor” “Julius Caesar was not the first Chinese emperor, and Peter the Great was not the last German emperor.” This is true.
1.1. LOGICAL CONNECTIVES 5
- The negation of “ 2 < 3 , or π is the area of a circle of radius 2” is “ 2 ≥ 3 , and π is not the area of a circle of radius 2.” This is false, because the first part is false. 1.1.6. The implication if...then****. Next, we consider the implication “if A, then B” or “A implies B.” We define “if A, then B” to mean “not(A and not(B)),” or, equivalently, “not(A) or B”; this is fair enough, since we want “if A, then B” to mean that one cannot have A without also having B. The negation of “A implies B” is thus “A and not(B)”.
Exercise 1.1.2. Write out the truth table for “A implies B” and for its negation.
Definition 1.1.1. The contrapositive of the implication “A implies B” is “not(B) implies not(A). ” The converse of the implication “A implies B” is “B implies A”.
The converse of a true implication may be either true or false. For ex- ample:
- The implication “If − 3 > 2 , then 9 > 4 ” is true. The converse im- plication “If 9 > 4 , then (− 3 ) > 2 ” is false.
However, the contrapositive of a true implication is always true, and the contrapositive of a false implication is always false, as is verified in Exer- cise 1.1.3.
Exercise 1.1.3. “A implies B” is equivalent to itscontrapositive “not(B) implies not(A).” Write out the truth tables to verify this.
Exercise 1.1.4. Sometimes students jump to the conclusion that “A implies B” is equivalent to one or another of the following: “A and B,” “B implies A,”, or “not(A) implies not(B).” Check that in fact “A implies B” is not equivalent to any of these by writing out the truth tables and noticing the differences.
Exercise 1.1.5. Verify that “A implies (B implies C)” is logically equiv- alent to “(A and B) implies C,” by use of truth tables.
Exercise 1.1.6. Verify that “A or B” is equivalent to ‘if ‘not(A), then B,” by writing out truth tables. (Often a statement of the form “A or B” is most conveniently proved by assuming A does not hold, and proving B.)
The use of the connectives “and,” and “not” in logic and mathematics coincide with their use in everyday language, and their meaning is clear. The use of “or” in mathematics differs only slightly from everyday use, in
1.2. QUANTIFIED STATEMENTS 7
1.2. Quantified statements
1.2.1. Quantifiers. One frequently makes statements in mathematics which assert that all the elements in some set have a certain property, or that there exists at least one element in the set with a certain property. For example:
- For every real number x , one has x^2 ≥ 0.
- For all lines L and M , if L 6 = M and L ∩ M is non-empty, then L ∩ M consists of exactly one point.
- There exists a positive real number whose square is 2.
- Let L be a line. Then there exist at least two points on L. Statements containing one of the phrases “for every”, “for all”, “for each”, etc. are said to have a universal quantifier. Such statements typi- cally have the form:
- For all x, P ( x ) ,
where P ( x ) is some assertion about x. The first two examples above have universal quantifiers. Statements containing one of the phrases “there exists,” “there is,” “one can find,” etc. are said to have an existential quantifier. Such statements typically have the form:
- There exists an x such that P ( x ) ,
where P ( x ) is some assertion about x. The third and fourth examples above contain existential quantifiers. One thing to watch out for in mathematical writing is the use of implicit universal quantifiers, which are usually coupled with implications. For ex- ample,
- If x is a non-zero real number, then x^2 is positive
actually means,
- For all real numbers x , if x 6 = 0, then x^2 is positive,
or
- For all non-zero real numbers x , the quantity x^2 is positive.
1.2.2. Negation of Quantified Statements. Let us consider how to form the negation of sentences containing quantifiers. The negation of the assertion that every x has a certain property is that some x does not have this property; thus the negation of
is
- There exists an x such that not P ( x ).
8 1. LOGIC AND SETS
For example the negation of the (true) statement
- For all non-zero real numbers x , the quantity x^2 is positive
is the (false) statement
- There exists a non-zero real numbers x , such that x^2 ≤ 0. Similarly the negation of a statement
- There exists an x such that P ( x ).
is
- For every x, not P ( x ). For example, the negation of the (true) statement
- There exists a real number x such that x^2 = 2_._
is the (false) statement
- For all real numbers x, x^2 6 = 2_._ In order to express complex ideas, it is quite common to string together several quantifiers. For example
- For every positive real number x, there exists a positive real number y such that y^2 = x.
- For every natural number m, there exists a natural number n such that n > m.
- For every pair of distinct points p and q, there exists exactly one line L such that L contains p and q.
All of these are true statements. There is a rather nice rule for negating such statements with chains of quantifiers: one runs through chain changing every universal quantifier to an existential quantifier, and every existential quantifier to a universal quan- tifier, and then one negates the assertion at the end. For example, the negation of the (true) sentence
- For every positive real number x, there exists a positive real number y such that y^2 = x.
is the (false) statement
- There exists a positive real number x such that for every positive real number y, one has y^2 6 = x.
1.2.3. Implicit universal quantifiers. Frequently “if ... then” sentences in mathematics also involve the universal quantifier “for every”.
- For every real number x, if x 6 = 0 , then x^2 > 0.
Quite often the quantifier is only implicitly present; in place of the sentence above, it is common to write
- If x is a non-zero real number, then x^2 > 0.
10 1. LOGIC AND SETS
and reverse the two quantifiers, you get the totally absurd statement:
- There exists a positive real number y such that for every positive real number x, one has y^2 = x.
1.2.5. Negation of complex sentences. Here is a summary of rules for negating statements:
- The negation of “A or B” is “not(A) and not(B).”
- The negation of “A and B” is “not(A) or not(B).”
- The negation of “For every x, P(x)” is “There exists x such that not(P(x)).”
- The negation of “There exists an x such that P(x)” is “For every x, not(P(x)).”
- The negation of “A implies B” is “A and not(B).”
- Many statements with implications have implicit universal quanti- fiers, and one must use the rule (3) for negating such sentences. The negation of a complex statement (one containing quantifiers or log- ical connectives) can be “simplified” step by step using the rules above, until it contains only negations of simple statements. For example, a state- ment of the form “For all x , if P ( x ), then Q ( x ) and R ( x )” has a negation which simplifies as follows:
not(For all x , if P ( x ), then Q ( x ) and R ( x )) ≡ There exists x such that not( if P ( x ), then Q ( x ) and R ( x )) ≡ There exists x such that P ( x ) and not( Q ( x ) and R ( x )) ≡ There exists x such that P ( x ) and not( Q ( x ) ) or not( R ( x ) ).
Let’s consider a special case of a statement of this form:
- For all real numbers x, if x < 0 , then x^3 < 0 and | x | = − x.
Here we have P ( x ) : x < 0, Q ( x ) : x^3 < 0 and R ( x ) : | x | = − x. Therefore the negation of the statement is:
- There exists a real number x such that x < 0 , and x^3 ≥ 0 or | x | 6 = − x. Here is another example
- If L and M are distinct lines with non-empty intersection, then the intersection of L and M consists of one point.
This sentence has an implicit universal quantifier and actually means:
- For every pair of lines L and M, if L and M are distinct and have non-empty intersection, then the intersection of L and M consists of one point.
Therefore the negation uses both the rule for negation of sentences with universal quantifiers, and the rule for negation of implications:
1.2. QUANTIFIED STATEMENTS 11
- There exists a pair of lines L and M such that L and M are distinct and have non-empty intersection, and the intersection does not con- sist of one point.
Finally, this can be rephrased as:
- There exists a pair of lines L and M such that L and M are distinct and have at least two points in their intersection.
Exercise 1.2.1. Form the negation of each of the following sentences. Simplify until the result contains negations only of simple sentences.
(a) Tonight I will go to a restaurant for dinner or to a movie. (b) Tonight I will go to a restaurant for dinner and to a movie. (c) If today is Tuesday, I have missed a deadline. (d) For all lines L , L has at least two points.
(e) For every line L and every plane P , if L is not a subset of P ,
then L ∩ P has at most one point.
Exercise 1.2.2. Same instructions as for the previous problem Watch out for implicit universal quantifiers.
(a) If x is a real number, then
x^2 = | x |. (b) If√ x is a natural number and x is not a perfect square, then x is irrational. (c) If n is a natural number, then there exists a natural number N such N > n. (d) If L and M are distinct lines, then either L and M do not intersect, or their intersection contains exactly one point.
1.2.6. Deductions. Logic concerns not only statements but also de- ductions. Basically there is only one rule of deduction:
- If A, then B. A. Therefore B.
For quantified statements this takes the form:
- For all x, if A ( x ) , then B ( x ). A (α). Therefore B (α).
Example:
- Every subgroup of an abelian group is normal. Z is an abelian group, and 3 Z is a subgroup. Therefore 3 Z is a normal subgroup of Z_._
If you don’t know what this means, it doesn’t matter: You don’t have to know what it means in order to appreciate its form. Here is another example of exactly the same form:
- Every car will eventually end up as a pile of rust. My brand new blue-green Miata is a car. Therefore it will eventually end up as a pile of rust.
1.3. SETS 13
Given finitely many sets, for example, five sets A , B , C , D , E , one sim- ilarly defines their intersection A ∩ B ∩ C ∩ D ∩ E to consist of those elements which are in all of the sets, and the union A ∪ B ∪ C ∪ D ∪ E to consist of those elements which are in at least one of the sets. There is a unique set with no elements at all which is called the empty set , or the null set and usually denoted ∅.
Proposition 1.3.1. The empty set is a subset of every set.
Proof. Given an arbitrary set A , we have to show that ∅ ⊆ A ; that is, for every element x ∈ ∅, one has x ∈ A. The negation of this statement is that there exists an element x ∈ ∅ such that x 6 ∈ A. But this negation is false, because there are no elements at all in ∅! So the original statement is true.
If the intersection of two sets is the empty set, we say that the sets are disjoint , or non-intersecting. Here is a small theorem concerning the properties of set operations.
Proposition 1.3.2. For all sets A , B , C,
(a) A ∪ A = A, and A ∩ A = A. (b) A ∪ B = B ∪ A, and A ∩ B = B ∩ A. (c) ( A ∪ B ) ∪ C = A ∪ B ∪ C = A ∪ ( B ∪ C ) , and ( A ∩ B ) ∩ C = A ∩ B ∩ C = A ∩ ( B ∩ C ). (d) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) , and A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ).
The proofs are just a matter of checking definitions. Given two sets A and B , we define the relative complement of B in A , denoted A \ B , to be the elements of A which are not contained in B. That is, A \ B = { x ∈ A : x 6 ∈ B }. In general, all the sets appearing in some particular mathematical dis- cussion are subsets of some “universal” set U ; for example, we might be discussing only subsets of the real numbers R. (However, there is no uni- versal set once and for all, for all mathematical discussions; the assumption of a “set of all sets” leads to contradictions.) It is customary and convenient
to use some special notation such as C ( B ) for the complement of B relative
to U , and to refer to C ( B ) = U \ B simply as the complement of B. (The
notation C ( B ) is not standard.)
Exercise 1.3.1. The sets A ∩ B and A \ B are disjoint and have union equal to A.
14 1. LOGIC AND SETS
Exercise 1.3.2 (de Morgan’s laws). For any sets A and B , one has:
C ( A ∪ B ) = C ( A ) ∩ C ( B ),
and
C ( A ∩ B ) = C ( A ) ∪ C ( B ).
Exercise 1.3.3. For any sets A and B , A \ B = A ∩ C ( B ).
Exercise 1.3.4. For any sets A and B ,
( A ∪ B ) \ ( A ∩ B ) = ( A \ B ) ∪ ( B \ A ).
1.3.2. Functions. We recall the notion of a function from A to B and some terminology regarding functions which is standard throughout math- ematics. A function f from A to B is a rule which gives for each element of a ∈ A an “outcome” in f ( a ) ∈ B. A is called the domain of the function, B the co-domain , f ( a ) is called the value of the function at a , and the set of all values, { f ( a ) : a ∈ A }, is called the range of the function. In general, the range is only a subset of B ; a function is said to be sur- jective , or onto , if its range is all of B ; that is, for each b ∈ B , there exists an a ∈ A , such that f ( a ) = b. Figure 1.3.1 exhibits a surjective function. Note that the statement that a function is surjective has to be expressed by a statement with a string of quantifiers.
Figure 1.3.1. A Surjection
A function f is said to be injective , or one-to-one , if for each two dis- tinct elements a and a ′^ in A , one has f ( a ) 6 = f ( a ′^ ). Equivalently, for all a , a ′^ ∈ A , if f ( a ) = f ( a ′^ ) then a = a ′. Figure 1.3.2 displays an injective and a non- injective function.
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Chapter 2
Elements of Geometry
2.1. First concepts
The geometry which we will study consists of a set S , called space.
The elements of the set are called points. Furthermore S has certain dis-
tinguished subsets called lines and planes. A little later we will introduce
other special types of subsets of S , for example, circles , triangles , spheres ,
etc. On the one hand, we want to picture these various types of subsets ac- cording to our usual conceptions of them: Lines, planes, and so forth are idealizations of objects known from experience of the physical world. For example, a line is an idealization of a piece of string stretched tightly be- tween two points. (But it is supposed to extend indefinitely in both direc- tions, and, of course, we do not have any direct physical experience with anything of indefinite extent.) Similarly a plane is supposed to be a flat surface, like a table-top, but also is supposed to extend indefinitely in all directions. (Sort of like Nebraska, but larger. Again, we don’t have any di- rect physical experience with flat surfaces of indefinite extent.) We want to use our intuition and experience with physical space to suggest the results which should hold true in our geometry, and to guide our assumptions. On the other hand, it is a fundamental goal of a logical treatment of ge- ometry to make all of our assumptions quite explicit. We want to try to be very careful not to use in any proof any hidden assumptions about geomet- ric objects. Only in this way can we be sure that our arguments are correct, and that we can trust our results. We will allow ourselves the use of the real numbers, and all of their usual properties.
Axiom I-1 Given two distinct points, there is exactly one line contain- ing them.
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2.1. FIRST CONCEPTS 17
A
B
Figure 2.1.1. Axiom I-
Remember, a line is a set of points, and containment here means con-
tainment as elements. We denote by ← PQ → the line containing distinct points P and Q We call any collection of points which lie on one line colinear and any collection of points which lie on one plane coplanar
Axiom I-2 Given three non-colinear points, there is exactly one plane containing them.
C
B
A
Figure 2.1.2. Axiom I-
Axiom I-3 If two distinct points lie in a plane P , then the line con-
taining them is a subset of P.
A B
Figure 2.1.3. Axiom I-
Axiom I-4 If two planes intersect, then their intersection is a line.
2.1. FIRST CONCEPTS 19
Figure 2.1.6. Intersection of a Line and a Plane
So far, all the axioms (and two theorems) would be valid for a geometry with only one point P with { P } begin both a line and an plane! So clearly the axioms so far do not force us to be talking about the geometry which we expect to talk about! Very shortly, I will give axioms which ensure that space has lots of points, but in the meanwhile let us at least assume the fol- lowing:
Axiom I-5 Every line has at least two points. Every plane has at least
3 non-colinear points. And S has at least 4 non-coplanar points.
Theorem 2.1.3. If L is a line, and P is a point not in L, then there is exactly
one plane P containing L ∪ { P }.
Proof. Exercise.
L P
Figure 2.1.7. Plane determined by a Line and a Point
20 2. ELEMENTS OF GEOMETRY
Theorem 2.1.4. If L and M are two distinct lines which intersect, then there is exactly one plane containing L ∪ M.
Proof. Exercise.
L
M
Figure 2.1.8. Plane determine by Two Lines
2.2. Distance
A familiar notion in geometry is that of distance. The distance between two points is the length of the line segment connecting them. In order to get things into logical order, we will actually introduce the notion of distance first, and use it to establish the notion of line segment!
Axiom D-1 For every pair of points A , B there is a number d ( A , B ) , called the distance from A to B. Distance satisfies the following proper- ties:
1. d ( A , B ) = d ( B , A ) 2. d ( A , B ) ≥ 0 , and d ( A , B ) = 0 if, and only if, A = B.
Definition 2.2.1. A coordinate function on a line L is a bijective (one-to- one and onto) function f from L to the real numbers R which satisfies | f ( A ) − f ( B )| = d ( A , B ) for all A , B ∈ L. Given a coordinate function f , the number f ( A ) is called the coordinate of the point A ∈ L.
Axiom D-2 Every line has at least one coordinate function.
It follows immediately that every line contains infinitely many points, because R is an infinite set, and a coordinate function is a one-to-one cor- respondence of the line with R. Any coordinate function makes a line into a “number line” or “ruler”.
