Semantic difference between Set and TypeSets and types have equivalent notions in their respective theories. They represent a way to build and talk about structured, complex mathematical objects, but the way they are used is quite different. These two terms play a vital role in terms of mathematics as they give rise to set and type theory. Both sets and types are sorts of containers that hold and categorize objects based on their properties. Sets can group many heterogeneous objects together based on their properties and are thus more flexible. On the other hand, types classify objects based on their construction which gives an idea for how to destruct an arbitrary member of a specific type. In this article, we will discuss the semantic difference between set and type in terms of mathematics. What are sets in mathematics?In mathematics, a set is defined as the collection of objects. The objects in a set are known as elements of the set. The elements in a set need to be clearly defined to know whether or not a given object is an element of the set. In a class of students, each student is an element of the class or set of students. Set was founded in the 19^{th} century, and George Cantor is known as the father of set theory.
Let's understand the set theory concept with the help of an example. If P = the set of whole numbers greater than 2 and less than 7, then P = {3,4,5,6} This is a clearly defined set as we have 3, 4, 5, 6 is greater than 2 and less than 7. Therefore, the set of all tall students in our class does not form a mathematical set as all elements of members are not clearly defined. Ideology of setThe ideology of the set is a "collection of objects." Consider predefined objects in mathematics. These predefined objects can be studied, analyzed, and can put together in different forms based on certain factors. These groups form a collection, and a collection of objects is a set. Applications of setSet has broad application in mathematics. Some most significant applications of the set are given below.
Examples based on a setExample 1: Find the solution set of the equation x^{2}  9 = 0 in roster form? Solution: Given; x^{2}  9 = 0 = x^{2}  3^{2} = 0 = (x + 3) (x  3); {a^{2}  b^{2} = (a + b) (a  b)} = 3, 3 Thus, A = {3, 3} Example 2: If P = {1,2,3,4,5,6,7} and Q = {2,5,7,8}, find PQ and QP Solution: P  Q = {1,3,4,6} Q P = {8} Here, P  Q is not equal to Q P What is Type?Type in mathematics is a type of collection of values that are produced on evaluation of the term, and type is denoted by T. Type leads the type theory. In other words, type Theory refers to a foundation of Mathematics and an alternative to set theory. Category theory is a very abstract field of Mathematics that is extremely useful to classify and generalize constructions. However, category theory and Type Theory work very well together, and most concepts in type theory can be briefly expressed in categorical terms.
Ideology of type:
Features of type:
Applications of TypeThe type has broad application in mathematics. Some significant applications of type are given below.
