Wednesday, January 16

Set theory



We deal with many types objects in mathematics and when we combine them and out them together, we call them Sets Math and study of that collection of objects is the theory of sets.

The objects of the sets are usually called elements.

Set Theory Notation– We know that the collection of elements is termed as sets. We usually write a st in curly brackets.
For example: - A st of first five even numbers can be written as {2, 4, 6, 8, and 10} or the saet of first seven natural numbers is written as {1, 2, 3, 4, 5, 6, 7}.

The intersection of two sets is symbolized by inverted U and the union of two sets is written as with symbol U. Union of sets A and B means the saet of all objects that is member of both the sets and this can be written as A U B.

The intersection of two sets that is A intersection B is all the elements that are part of both A and B. If A is contained or part of B than we can call A as subset of B or if B has A contained in it then we can call B as superset of A.

We use different types of axioms in the theory of sets. Axioms can be logical or non-logical. Logical axioms tells us the universal truth which is like a + b = b + a.

An axiom act as a starter to derive the proofs similarly in Set Theory Proofs, we make use of Axiomatic Set Theoryto derive various proofs. 

There are many types of axioms that are used in the axiomatic theory which are axiom of existence, axiom of equality, axiom of pair, axiom of separation, axiom of union, axiom of power, axiom of infinity, axiom of image, axiom of foundation and axiom of choice.

Set Theory Problems are simplified using the axioms of the theory.

We have different types of special sets that we use in mathematics and that are used in the theory of sets.
For example the saet of all integers is denoted by Z where Z = {…-3, -2, -1, 0, 1, 2, 3…} and st of all prime numbers is denoted by P where P = {2, 3, 5, 7, 11, 13, 17, 19, 23…}

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