functions of real variable and functional analysis study notes[1]

Mapping and Potentiality

1. Every infinite set is equinumerous to some proper subset of itself.
2. a set equaled the set $\mathbb{N}$ consisting of all natural numbers is countable set,also called countably infinite.
3. The set of real numbers between 0 and 1 is uncountable.
4. the cardinality of [0,1] (and hence R) is strictly greater than that of N.

the relation of equivalence

1. there is a assumption that A is a set and the exists certainly a following relation between each element of A .

• reflexivity: $a\sim a$ for all $a\in A$
• symmetry: if $a\sim b$,then $b\sim a$ $(a,b)\in A$
• transitivity: if $a\sim b,b\sim c$,then $a\sim c$

2. let A is defined as a set, { A i , i ∈ I } \{A_i,i \in I\} {Ai​,i∈I}are a group of subset of A .
   if $A_i\cap A_j=\varnothing (i\ne j)$ and ${\cup }_{i\in I}{A}_i=A$, the { A i , i ∈ I } \{A_i,i \in I\} {Ai​,i∈I} is identified as a division of A.

references

1 .《实变函数论与泛函分析》