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Semantic difference between Set and Type

Sets 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 19th century, and George Cantor is known as the father of set theory.

{} Set It denotes a collection of elements
A ꓴ B A union B The elements that belong to set A and Set B
A Ⴖ B A intersection B The elements that belong to set A and Set B
A ⊂ B Proper subset A is a subset of B, but A is not equal to B
A ⊆ B Subset A is a subset of B, and Set A is included in set B
A ⊄ B Not Subset Set A is not a subset of Set B

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 set

The ideology of the set is a "collection of objects." Consider pre-defined objects in mathematics. These pre-defined 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 set

Set has broad application in mathematics. Some most significant applications of the set are given below.

  • A set is used to construct relations.
  • Set is used to identify and store unique elements in computer programming.
  • Set is used in the most important branch of computer science that is digital electronics.
  • Set is the foundation of Boolean algebra.
  • Set is the basic foundation of set theory.

Examples based on a set

Example 1:

Find the solution set of the equation x2 - 9 = 0 in roster form?



x2 - 9 = 0

= x2 - 32 = 0

= (x + 3) (x - 3); {a2 - b2 = (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 P-Q and Q-P


P - Q = {1,3,4,6}

Q- P = {8}


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.

True 1
False 0
A and B A×B
A or B A+B
If A then B A→B
A if and only if B A→B × B→A
Not A A→0

Ideology of type:

  • Construction is the ideology of type.
  • In mathematics, the objects are constructed as per the rules.
  • The organization of different objects based on their construction categorizes them into different types.
  • In mathematics, objects are constructed uniquely, resulting in unique types.

Features of type:

  • The type has "Terms" included with it
  • , and the terms belong to only one type in the type theory.
  • Numbers are used to representing functions in type theory. Terms "and" and "or" can be encoded as themselves. No, a particular system is required.
  • Features included with type are dependent types, inductive types, universe types, equality types, and computational components.
  • If you want to check a term of a particular type, "algorithm is needed."

Applications of Type

The type has broad application in mathematics. Some significant applications of type are given below.

  • Type theory is used in the semantics of the language.
  • In programming languages, type systems are used to identify bugs.
  • Gregory Bateson logical level and Double bind notions are derived from type theory.
  • Proof theorem, proof checkers, and proof assistant use type theory of encoding proofs in mathematics.
  • Type theory creates practical implementation, programming languages, mathematical foundations, and social sciences.

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