Set solution

Elementary Set Theory and Relational Propositions.

Background Analysis.

The logicist programme of A.N. Whitehead and Bertrand Russell reached its zenith in Principia Mathematica, a work of immense significance in the domain of symbolic logic and pure mathematics. The book is a thorough and ingenuous expose of pure mathematics with a few primitive propositions of logic. In the work, the authors discussed extensively different kinds of relations which form a common ground of theoritical disquisitions amongst logicians and mathematicians. Additionally, Principia Mathematheca elucidated the theory of classes and the relations that may obtain between them. In order to appreciate fully the ground-breaking achievements of that work, it is necessary to remember that before its publication the theory of numbers and the calculus of relations had floated for quite some time in the thick fog of metaphysics and mysticism. For example, Aristotle's theory of substance and essences obfuscated the distinction between names and predicates and consequently engendered untenable theories about relations. Leibniz, whose work on mathematical logic would have superseded Aristle's logic even made mistakes in his analysis of infinite collections or sets. He had noticed that the number of even numbers must be equal to the number of all whole numbers, and thought it was a contradiction. The mathematician, George Cantor rejected Leibniz's position and posited a better theory of numbers. Again, Hegel's amalgamation of logic with methaphysics was very inimical to the growth of the former. Infact, Hegelians either ignored relations altogether or advanced specious arguments which purport to prove the unreality of relations.

Russel and Whitehead provided a mathematical framework for the detail spelling out of relations and classes amongst others. The theoritical framework necessitated new algorithms which made it possible to treat symbolically many technical questions previously left to the fuzziness and nebulosity of ordinary language. This is particularly true in the field of relations.

In this chapter we shall consider some of the elementary principles and methods employed by logicians and mathematical logicians in the analyses of sets and relational propositions.

Sets: Membership and Operations on Sets

The idea of grouping things into bundles, aggregates, or sets is a very common one. It is presupposed when we collect a bundle of wirewood, or when someone thinks about the collection of books in a university library. Even, a bag of groundnuts or a bunch of plantain constitute a set. Generally speaking, the word "set" is used to refer to any kind of collection of things of any sort. Individual entities that make up a set are called element or members of that set. In most cases, it is possible for us to say with regard to any entity and any set that a particular entity is an element of that set or it is not. Thus, an apple is not a number of the set of all insects, a butterfly is. The square-root of the number 2 is not a member of the set of integers, but the number 2 belongs to that set.

It is customary, in the discussion of sets, to commence with a definite class of objects on which the analysis is focused. This is referred to as the universal set. The universal or basic set E varies from one problem to another. Sometimes, for example, we may focus our attention on the positive integers, or on the set of legislators, etc.

Usually, a set can be defined in two ways. One, we may enumerate all its members and then enclose them with braces. This is the roster method. The set of even numbers less than 14 is {2, 4, 6, 8, 10, 12}. Ellipses can occur within the braces if and only if one can tell precisely what and what are the intervening entities whose spaces are taken up by the ellipses. The expression {1, 2, 3, 4,...50} is the set of positive integers from 1 to 50. But we cannot define an infinite collection or a collection whose numbers are unknown in this way. So we introduce the second method in which a set is defined by stating the property or properties that must be possessed by each and every member of the set. The second definition is more popular amongst scholars. It is based on the principle of abstraction, namely, that every property defines a set. All the alphabets in Igbo language, for instance, constitute a set. So do all the fingers in a hand. The sets we have discussed so far are finite. Of course, there are infinite sets, that is sets that contain an infinite number of elements. The set of natural numbers is a good example. Given a basic set, we can talk about its subset. A subset s of t is a set whose elements are all members of another set. The philosophy lecturers in the University of Boston form a subset of all human beings. Again, the footballs used by FIFA in last year's world cup tournament are a subset of all footballs. Certainly a subset, just like a set, does not have to be tangible; infact sets and subsets are generally intangibles. In mathematical logic, for example, the odd integers are a subset of integers, both of which are intangibles. Assuming ther are two set s and t such that every member of s is a member of t also, we say that s is included in t and that s is a subset of t. Symbolically the relation is written as

s⊂t

If we wish to affirm that s is not a subset of t, we symbolize it thus

s⊆t

For instance, the set of motor cars is not a subset of the set comprising aeroplanes. It may happen that both s⊂t and t⊂s are equal, meaning that the two sets have exactly identical numbers. This implies that s and t are equal. The equality of sets is referred to as the principle of existensionlity. For instance, the sets s = {A, B, C, D} and t = {D, A, C, B} are equal inspite of the fact that the elements are arranged differently in each set.

If it is stated that s⊆t and t⊂s, s is said to be a proper subset of t. It is written sd.

s⊂t

There are some basic operations that can be performed on sets. such operations are generally in agreement with the Boolean interpretation of classes.

The first operation we shall consider is the union of sets. The union-set of two sets s and t is the set of objects which belong either to s or to t. A collection of students may belong to the philosophy department or to mathematics. The union of s and t is designated as:

........ s ∪ t The intersection-set or product of s and t is the set of all those objects that are elements or members of both s and t. The intersection of these sets is symbolically stated in this manner.
........ s ∩ t The complement of a set s is made up of those objects that do not belong to s. It is rendered in symbols as S (read "s bar") In mathemathics and mathematical logic, a set is exemplified usually by an imaginary number, indicating clearly that sets and subsets are intangible. Generally, a subset is a set in its own right, and the number of logically possible sets in infinite. A property that delineates a set does not have to be tangible even when the objects to which it applies are tangible. To illustrate this point, I may be thinking about the set of all the triangles in a concrete pyramid, but triangularity is not a tangle property. Now, we express the relation between any individual x, that is an element of s, as x ∈ s The symbol ∈ is the Greek letter epsilon, and symbolizes the relation of being "a member of or is an element in" a set. AS an illustration, supposing that s represents the set of "fairly-used cars" and t symbolizes the set of "cars" then s is a subset of t and any member of s is also a member of t. This is plain enough, since every fairly-used car is a car. Logically, the relation involved here can be generalized as (x) : (x ∈ A) ⊂ (x ∈ B) Since a set may have any number of elements, it may also have just a single number. A set with only one element is called a unit set. In set theory, we distinguish the element 1 and the set {1}. The symbol {1}, is the unit set of the number 1. Thus, it is the set whose only member is the element 1. in general, given any indidual a, we may define ia, the unit set of a, as: ia= {(x ∈ v) | x= a} This follows from the generalization of the principle of abstraction which is written thus x { ...|x ...} and read as "the set of all x's such that..." There is an important set in mathematical logic which is usually neglected in ordinary discussion. It is the set with no elements at all. We call such a set the null or empty set, and in the notation of Boolean algebra, it is symbolised as Ø. Accordingly, if there is no x we then have x ∈ Ø. A radical difference exists between ∈ on the one hand and ⊆ or ⊂, on the other. To begin with,⊆ and ⊂ (together with ∪,∩ and") apply to sets alone. When we affirm that the set whose numbers are footballers is included in the set of athletes (using s and t to substitute for the set of footballers and the set of athletes respectively), we then write, using the usual notation, s ⊂ t So, the relation of inclusion cannot hold between elements of sets or between elements and sets. It is possible to state that {1,2} ⊂ {1,2,3,4,} but never I ⊂ I nor I ⊂ {1,2} Similarly since the relation ∈ holds between an element and the corresponding set, we cannot in general assert of an element x that x ∈ x but we may say x ∈ s it is important to note that sets can be members of other sets, as in {1,2}3} and {1},2}3} It does not really matter the order in which the elements in a set are written down. The following are different arrangements of four alphabets which constitute a set: {q, r, s, t} = {t, s, r, q} = {r, q, s, t} = {r, t, q, s} ... and so on. The number of elements which a set contains is called the cardinality of that set. In our example, the cardinality of the set is four. By definition, a set s contains each of its element exactly once and no repetitions are permitted. Its subsets, as we said earlier, are those sets whose elements are contained in s. To determine the number of subses in the universal set E, we must include the set itself and the null set as the subsets of E also. Given any set whatsoever, the number of its subsets is derivable from the formula N = 2n, where N represents the number of its subsets in s. As an illustration consider the number of students who distinguished themselves in a logic examination. These students (let us call them S1, S2S3) constitute the universal set. The number of possible subsets we can derive from the situation are {S1,S2,S3,},{S1,S2,},{S1,S3,}, {S2,S3,}, {S2,S3,},{S1,},{S2,},{S3,} and {o}. This follows from N = 2n. Note also that here both the universal set and the null set are included as subsets of the universal set. A set with a single member, as you can see, has 2 possible subsets. There is an important method for forming sets which is different from unions and intersections. This is the method whereby sets are constructed from ordered pairs. An ordered pair is made up of two elements taken in a specified order; for example a and b in that order. Thus, to assert that (a, b) is an ordered pair, we intend to say that a comes before b or, (a, b) ≠ (b, a) When we say that "Abel comes before cane in the alphabetical order", we are saying something totally different from "Cane comes before Abel in the alphabetical order". Now, taking a and b as arbitrary non empty sets, we define the Cartesian product a x b of these two sets as the ensemble of all ordered pairs which can be constructed by taking a first term from a and a second one from b. From the definition, it follows that if {a,b} is an ordered pair, we can affirm, in relation to another ordered pair, that there is a one-to-one correspondence between their elements. Thus, if the second ordered pair b {1,2} then {a,b} = {1,2} if and only if a = 1 and b = 2. But {a,b} ≠ {1,2} Also, a x b is greater than (>) a or b since if a has x elements and b has y elements then a x b has xy elements. We shall try an example to illustrate this point.

Let a = {0, 1} and b = {0, 2} we then have

a x b = {(0, 0),(0,2),(4,0),(1,2)}

b x a = {(0,0),(0,1),(2,0),(2,1)}

Given a universal set E some relations may link pairs of elements of E in a particular sequence. Take the relation of "taller than", it selects precisely those pairs of individuals actually related in such a manner that the first is taller than the second. Assume futher than a set constructed with the relation "taller" than has just four elements. By using the cartesian product in a tabular form, we can construct all possible pairs of elements of the set s. Labelling the four elements 1,2,3 and 4 respectively, the set in question can generate sixteen ordered pairs:

4 (4,1), (4,2), (4,3), (4,4)

3 (3,1), (3,2), (3,3), (3,4)

2 (2,1), (2,2), (2,3), (2,4)

1 (1,1), (1,2), (1,3), (1,4)

1 2 3 4

Our table depicts all possible ordered pairs that can be generated from the relation of "taller man" in a set of four elements. The same principle can be used to generate ordered pairs for finite and infinite sets. In the case of an infinite set is it enough to write down the first few pairs as illustrations, followed by three dotted lines to indicate that the list go on and on.

To instaciate what can be referred to as ordered "4-tuples" the following diagram is quite representative.

Consider a logician who takes gambling as a hobby. In each game he played on a particular occasion he has two different options. Let the two options be a set of elements A and B. If the logician has to play four times, we calculate the possibilities before him as follows:

B,B | A,A,B,B,| A,B,B,B, | B,A,B,B, | B,B,B,B, |

B,A | A,A,B,A,| A,B,B,A, | B,A,B,A, | B,B,B,A, |

A,B | A,A,A,B,| A,B,A,B, | B,A,A,B, | B,B,A,B, |

A,A | A,A,A,A,| A,B,A,A, | B,A,A,A, | B,B,A,A, |

A,A A,B B,A B,B

The table completely exhausts all the possibilities available within the logic of the situation. In otherwords, we have generated sixteen ordered 4-tuples. Applying the expression to the ordered pairs we constructed initially, we refer to them as 2-tuples.

Propositions of predicate logic can be translated into set-theoretical symbols without any loss in meaning. Since any predicate or property determines a set, a monadic formula such as f(x), where the possession of f determines a set s, can be rewritten as x s. Thus since if in f(x1...x2) determines s we can rewrite the whole formula as (x1...x2) s. Logicians such as Russell and whitehead have fruitfully exploited the ramifications of set theory into other domains to set up an expressive theory of relations.

Elementary Postulates of the Algebra of Sets.

Following upon what we stated about the translatability of predicate logic into set-theoretic terms, some basic postulates of set theory clearly resemble the rules of inference in propositional logic. We shall list fourteen of these postulates and illustrate a few of them with venn diagram. But before we go into all that, two other operations on sets have to be introduced. The two operations are difference and exclusive union.

Let s and t be any two subsets of the universal set E. The difference between s and t, written as s-t, is the subset that consists of those elements which are members of s but are not numbers of t. The difference s-t is clearly exhibited in the diagram below.

The exclusive union of s and t, denoted by sUt is the set of elements of the universal set that belong to s or t but not both. The following venn Diagram illustrates sUt.

On closer inspection, it can be observed that s-t and sUt can be derived from the three operations on sets namely union, intersection and complementation.

Thus s-t can be expressed as s n t, while 5 u t can be penned down as (s n t) n (s u t) or as (t n s) u (s n t). Returning now to the postulates we undumbrated earlier, a few of interest to the student of Symbolic logic. They include:

Postulates 10 down to 12, rely on the definition of the empty set Q universal set E and the complement set. They are exhibited clearly in this Venn Diagram:

This diagram illustrates clearly that the intersection of the set s and the universal E (that is, S n E) is precisely covered by the area enclosed within the circle. This This diagram illustrates clearly that the
intersection of the set s and the universal E (that
is, S n E) is precisely covered by the area enclosed
within the circle. This establishes postulate 10.
Further, the union of s and 5 includes the space
within the circle and the space outside the circle
enclosed in the boundaries of ABCD, that is within
the universal set E. But these spaces are all within
E. Therefore, S U S equals E. Finally, S is the
complement set of S. Our diagram does not
contain any space corresponding to the intersection of
f s and its complementary set. This means that s
intersection 5 represents an empty or null set. The
null set, you would recall, has no members at all.
From the diagram then, it follows that nothing
belong to both 5 and s. The use of the set algebra
to illustrate sets is demonstrated further in the
Venn Diagram below:

This ﬁgure represents the
universal set and two subsets within it. Different
spaces in the diagram are demarked by the letters
a, b, c and u. Those areas or spaces consisting of a, b, c, u make up the universal set E. The set 5
consists of a and b whereas b and c make up set t.
All the possible subsets contained in the Venn
Diagram can be constructed using the operations
on sets earlier introduced:

SetSpace
as n t v s - t
bs
nt
csntvt-s
u (sUt)
a, bs
a,c(snt) U (s
nt)n(sUt),v(sUt),v(s-t)U (t-s)
a,ut
b,ct
c,u s
a, b, cs U t
a, c, u (s n t)
a, b, c, u E

Example
1Determine w

which of the following statements are true and
which are false.

(a) D E {A, B, C, E, F}

(b) A E {1, 2, 3, 4}
(c) F E {2, 4, 6, 8,
10
(d) {1}E]

Solution

(a)
False (b) False
(c) True
(d) False

Example
2

(a) {1,2, 3, 4} U {2, 3, 5} =
(b) {1, 2, 3,
4} n {4, 5, 3} =
(c) {5, 6, 7, 8} U {2, 3, 5} =
(d) {2, 4,
6, 8} n {5, 6, 7} =
(e) {1, 2, 5, 6} n {A, B, C}
=

Solution

br>
(a) 1, 2, 3, 4, 5 (b) 3,
4 (c) 2, 3, 5, 6, 7, 8
(d)
6 (e) Q

Example
3

Enumerate the subsets of 1, 2, 3,
4, 5 which have exactly two numbers.

1,
2, 1, 3, 1, 4, 1, 5, 2,3, 2, 4, 2, 5, 3, 4, 3,
5

Example 4

(a) Emumerate the
subsets of 1, 2, 3
(b) Which of them are
proper subsets of 1, 2, 3

So|ution

(a) 1, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 3, Q
(b) 1, 2, 3, 1, 2, 1, 3, 2, 3, Q

Example
5

For the sets 5 = a, b, c, d and t = b, d, e,
determine the membership of s U t, s n t, s U s, s n
Q, s U Q,

Solution

s U t = a, b, c,
d,e;snt=b,d;sUs=s;sns=s;an=Q;sU
Q: s.

Relations

Although
relational propositions could be interpreted in set

theoretical terms, there are well - developed alternatives for articulating and symbolizing such propositions. Relational propositions differ from subject-predicate propositions in that the latter links a subject and predicate together through a copula whereas as the latter does not. Really available illustrations of relational propositions are contained in phrases "to the east of", "on top of", "greater than" etc. In the statement "Obi is older than Ada", or "Aba is to the east of Owerri", Obi or Aba is the referent, and Ada or Owerri is the relatum. The relation between two individuals or objects as in the examples above is called a dyadic relation. In "A PDP Faithful gave one hundred and twenty

million naira to Obasanjo", we have a Relationship

between PDP faithful, one hundred and twenty
million naira and Obasanjo. We call such a relation
between three entities a triadic relation. An
example of tetradic relation is contained in the
proposition "Enwerem insisted that the letter referred to in magazine was rejected by a competent court of law,’I

Relations can be characterized in other
ways. For instance, from the proposition "Unoka is
the father of Okonkwo", it is not possible to infer
that "Okonkwo Is the father of Unoka". Thus the
relation "father of" is asymmetrical. But in "One
hundred kobo are equal to one naira," the
relation of equality asserted is symmetrical, since
a = b is the same as b = 3. Therefore we can infer
from the given proposition the statement "One
naira is equal to one hundred kobo.‘I A
relationship like "loving" is somet sometimes symmetrical and sometimes
asymmetrical. We say that kind of relation is not
symmetrical.
Transitive relation is
illustrated in the following: If A is bigger than B
and B is bigger than C, then A is bigger than C. The
relation "bigger than" is transitive. But even if it is
true that A is the father of B and B is the father of
C, it does not follow that A is the father of C. Here
the relation is intransitive. Again, some relations
may be transitive in some contexts, and
intransitive in o others. They are called nontransltlve relations. A
good example of such a relation is "being a friend
of."
From the foregoing given any
relational proposition whatsoever, it appears that
some individuals or objects stand in some relation
to other individuals or objects.

Symbolizing Relational
Propositions

ln Symbolizing
propositions containing relational words, phrases
and expressions, the use of variables to replace
individuals is complemented with symbols for

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