Question

What is today's label for postulates and common notion?

Answer 1

Todays level of Postulates are as follow :

A postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasons.

The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or questions.

When used in the latter sense, "postulate", and "assumption" may be used interchangeably. In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., parallel postulate in Geometry.

Todays level of common notation are as follow:

Common notations in the field of mathematics are as follow :

1) The set of natural numbers 1, 2, 3, … and so on. It is a matter of style whether or not to include 0 in(N)

2) The set of positive and negative whole numbers, …,−2, −1, 0, 1, 2, …(Z)

3) The rational numbers, e.g. the numbers which can be represented as a fraction of two integers where the denominator is nonzero.(Q)

R: All real numbers.

Empty Set: The unique set with no elements, denoted by ∅.

Set Builder Notation: If P(x) is some statement about elements of a set Y, then {x∈Y:P(x)} is the set containing all objects x∈Y that satisfy P(x). As an example, if P(x) says “x is even”, then {x∈N:P(x)} is the set of even natural numbers.

Subset: If X and Y are sets, then Y is a subset of X, written Y⊆X, if everything in Y is in X.

Elements: If X is a set and a is something inside the collection represented by X, we say a is an element of X and write a∈X.

Power Set: If X is a set, then the power set of X, written P(X), is the set whose elements are exactly the subsets of X.

Union: If A and B are sets, then A∪B denotes their union, the set of all elements in at least one of A or B. If T is a set of sets, then ⋃A∈TA denotes the union of all the elements of T, i.e. a∈⋃A∈TA if and only if a∈A for some A∈T.

Intersection: If A and B are sets, then A∩B denotes their intersection, the set of all elements in bothA and B. If T is a set of sets, then ⋂A∈TA denotes the intersection of all the elements of T, i.e. a∈⋂A∈TA if and only if a∈A for all A∈T.

Set Equality: Two sets are considered equal if they have the same elements. That is, if every element of A is in B and every element of B is in A, then A=B. Equal sets are indistinguishable, so if A=B and A∈C, then B∈C.

Function: A function f:X→Y is a set of pairs (x,y) such that x∈X, y∈Y, and for every x∈X there is a unique y∈Y such that (x,y)∈f. If (x,y)∈f, then we say f(x)=y. Note that the y only needs to be unique for a fixed x, we can choose the same y for multiple x’s but only one y per x.

Inverse Image: If f:X→Y is a function and A⊆Y, then f−1(A)={x∈X:f(x)∈A}.

Many others notations are also used .

Ask for any doubts do like if u find it useful.