Set theory is a branch of mathematical logic that deals with the study of sets, which are collections of distinct objects considered as a single entity. It provides a foundation for other branches of mathematics, including algebra, topology, and analysis. Here’s a brief introduction to some fundamental concepts in set theory:

1. Sets: A set is a collection of distinct objects, called elements, which are considered as a single entity. Sets are usually denoted by curly braces {}, and elements are listed within the braces, separated by commas. For example:
   • $A=\{1,2,3,4\}$
   • $B=\{a,b,c\}$

   • C={x∣x is an even integer}

2. Cardinality: The cardinality of a set is the number of elements it contains. It is denoted by
   
   $\mathrm{<br>}$

   ∣A∣ or card(A)

    

   card(A). For finite sets, the cardinality is simply the counting of elements. For example, if
   $A=\{1,2,3,4\}$

    

   A={1,2,3,4}, then
   $\mathrm{\mid }\mathrm{<span\; style="text-transform:\; math-auto;">A<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;">\mid </span></span>}=4$

    

   ∣A∣=4. For infinite sets, the concept is more subtle, and we use different methods to compare their sizes.

3. Finite Sets and Infinite Sets: A set is called finite if its elements can be counted and there are a specific number of them. Otherwise, it is called infinite. For example, the set of natural numbers
   
   $\mathbb{}$

   N={1,2,3,…} is infinite because it has no last element.

4. Equality of Sets: Two sets are equal if and only if they have exactly the same elements. Symbolically,
   
   $$

   A=B if and only if for every element
   $\mathrm{<br>}$

   x∈A if and only if 

   x∈B.

5. Subsets: A set
   
   $$

   A is a subset of another set
   $$

   B if every element of
   $$

   A is also an element of
   $\mathrm{<br>}$

    

   B. Symbolically, we write
   $$

   A⊆B. If there exists at least one element in
   $$

   B which is not in
   $$

   A, then
   $$

   A is a proper subset of
   $$

   B, denoted
   $$

   A⊂B.

6. Power Set: For any set
   
   $\mathrm{<br>}$

    

   A, the power set of
   $\mathrm{<br>}$

    

   A, denoted
   $\mathcal{}$

   P(A), is the set of all subsets of
   $\mathrm{<br>}$

    

   A, including the empty set and
   $$

   A itself.

7. Cardinal Numbers: Cardinal numbers are a generalization of the natural numbers used to denote the sizes of sets. Each cardinal number represents an equivalence class of sets under the relation of equinumerosity (having the same cardinality). The smallest infinite cardinal is denoted by
   
   ${\mathrm{\aleph }}_0$

    

   ℵ0​ (aleph-null), representing the cardinality of the set of natural numbers.

These are just some basic concepts in set theory. It’s a rich and deep field with many more advanced topics such as ordinal numbers, transfinite numbers, cardinal arithmetic, and more.