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Cantor’s Concept of a set
A set is any collection of definite, distinguishable objects of our intuition or of our intellect to be conceived as a whole. The objects are called the elements or members of set
The intuitive principle of extension for sets
Two sets are equal if and only if (iff) they have the same members. i.e., .
The intuitive principle of abstraction
A formula (syn: property) defines a set by the convention that the members of are exactly those objects such that is a true statement. .
Operations with/for sets
- Union (Sum or Join)
- Intersection (Product or Meet)
- Disjoint Sets and are disjoint sets iff and they intersect iff
- Partition of Sets A partition of a set is a disjoint collection
of non-empty and distinct subsets of such that each member of is a member of some (and hence exactly one) member of .
For example: is a partition of .
- Absolute Complement of a set is usually represented by where is universal set.
- Relative Complement of a set is given by .
Theorems on Sets
- Self-dual Property: If and
- Self Dual:
- Idempotent Law:
- Idempotent Law:
- Absorption Law:
- Absorption Law:
- de Morgen Law:
- de Morgen Law:
The following statements about set A and set B are equivalent to one another
Function is a relation such that no two distinct members have the same first co-ordinate in its graph. is a function iff
- The members of are ordered pairs.
- If ordered pairs and are members of , then
- Other words used as synonyms for the word ‘function’ are ‘transformation’, ‘map’, ‘mapping’, ‘correspondence’ and ‘operator’.
Notations for functions
A function is usually defined as ordered-pairs, see above, and so that is (was) a way to represent where is an argument of and is image (value) of .
Other popular notations for are: , , , .
Intuitive law of extension for Functions
Two sets and are equal iff they have the same members (here, Domain and Range)
A function is into iff the range of is a subset of . i.e.,
A function is onto iff the range of is . i.e.,
- Generally a mapping is represented by .
A function is called one-to-one if it maps distinct elements onto distinct elements.
A function is one-to-one iff and
Restriction of Function
If and if , then is a function on , called the restriction of to and is usually abbrevated by .
Extension of function
The function is an extension of a function iff .
Let be the set of rational numbers. It is well known that is an ordered field and also the set is equipped with a relation called “less than” which is an order relation. Between two rational numbers there exists infinite number of elements of . Thus the system of rational numbers seems to be dense and so apparently complete. But it is quite easy to show that there exist some numbers (?) (e.g., etc.) which are not rational. For example, let we have to prove that is not a rational number or in other words, there exist no rational number whose square is 2. To do that if possible, purpose that is a rational number. Then according to the definition of rational numbers , where p & q are relatively prime integers. Hence, or . This implies that p is even. Let , then or . Thus is also even if 2 is rational. But since both are even, they are not relatively prime, which is a contradiction. Hence is not a rational number and the proof is complete. Similarly we can prove that why other irrational numbers are not rational. From this proof, it is clear that the set is not complete and dense and that there are some gaps between the rational numbers in form of irrational numbers. This remark shows the necessity of forming a more comprehensive system of numbers other that the system of rational number. The elements of this extended set will be called a real number. The following three approaches have been made for defining a real number.
- Dedekind’s Theory
- Cantor’s Theory
- Method of Decimal Representation
The method known as Dedekind’s Theory will be discussed in this not, which is due to R. Dedekind (1831-1916). To discuss this theory we need the following definitions:
Ordered Field: Here, is, an algebraic structure on which the operations of addition, subtraction, multiplication & division by a non-zero number can be carried out.
Dedekind’s Section (Cut) of the Set of All the Rational Numbers
Since the set of rational numbers is an ordered field, we may consider the rational numbers to be arranged in order on straight line from left to right. Now if we cut this line by some point , then the set of rational numbers is divided into two classes and . The rational numbers on the left, i.e. the rational numbers less than the number corresponding to the point of cut are all in and the rational numbers on the right, i.e. The rational number greater than the point are all in . If the point is not a rational number then every rational number either belongs to or . But if is a rational number, then it may be considered as an element of .
Let satisfying the following conditions:
Let . Then the ordered pair is called a section or a cut of the set of rational numbers. This section of the set of rational numbers is called a real number.
Notation: The set of real numbers is denote by .
Let then and are called Lower and Upper Class of respectively. These classes will be denoted by and respectively.