A ∩ B Intersection: in both A and B C ∩ D = {3, 4}
A ⊆ B Subset: every element of A is in B. {3, 4, 5} ⊆ D
A ⊂ B Proper Subset: every element of A is in B,
but B has more elements. {3, 5} ⊂ D
A ⊄ B Not a Subset: A is not a subset of B {1, 6} ⊄ C
A ⊇ B Superset: A has same elements as B, or more {1, 2, 3} ⊇ {1, 2, 3}
A ⊃ B Proper Superset: A has B's elements and more {1, 2, 3, 4} ⊃ {1, 2, 3}
A ⊅ B Not a Superset: A is not a superset of B {1, 2, 6} ⊅ {1, 9}
Ac Complement: elements not in A Dc = {1, 2, 6, 7}
When set universal = {1, 2, 3, 4, 5, 6, 7}
A − B Difference: in A but not in B {1, 2, 3, 4} − {3, 4} = {1, 2}
a ∈ A Element of: a is in A 3 ∈ {1, 2, 3, 4}
b ∉ A Not element of: b is not in A 6 ∉ {1, 2, 3, 4}
Ø
A ∪ B
union
Elements that belong to set A or set B
A ∩ B
intersection
Elements that belong to both the sets, A and B
A ⊆ B
subset
subset has few or all elements equal to the set
A ⊄ B
not subset
left set is not a subset of right set
Mike when are you going the us?
This text is italic
This text is important
This text is bold
Emphasized text
Mark text
Smaller text
Inserted text
Subscript
Superscript
This is man.
This is Jane.
This is a car.
This is a bus.
This is a train.
This is a cap.
Centered.
Centered.
Centered.
This is a Pen.
my pentext.
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 Leibniz, whose work on mathematical logic would have superseded Aristle's logic even made mistakes in his analysis of infinite collections or sets.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
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