# Question about set theory and ordered pairs

• reb659
In summary, the conversation discusses the set theoretic definition of an ordered pair in comparison to another proposed definition. The standard definition, (x,y)={{x},{x,y}}, can be shortened to {x,{x,y}} and allows for determining the first and second elements using only set theory. The alternative definition, (x,y)={x,{y}}, does not work due to the possibility of counterexamples.
reb659
Hi

I was reading through a textbook and I came across the set theoretic definition of an ordered pair (Kuratowski), where (x,y)={{x},{x,y}}, which apparently can be shortened to {x,{x,y}}. This seems to be the standard definition for an ordered pair in set theory so that we can determine both the first and second element using only set theory and no notions of "first" or "second". However, I am wondering why the less complicated definition (x,y)={x,{y}} does not also work.

Can anyone enlighten me?

If {x,{y}}={a,{b}} then either x=a and {y}={b} which is what we want, or
x={b} and a={y}. So if we just pick y and b arbitrarily we can come up with counterexamples to what an ordered pair should satisfy. So for example ({0},1)=({1},0) under your proposed definition

## 1. What is set theory?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects or elements. It provides a framework for understanding how these sets relate to each other and how they can be manipulated and analyzed.

## 2. What is an ordered pair?

An ordered pair is a set of two elements, arranged in a specific order. It is commonly denoted as (x,y), where x and y are the two elements in the pair. The order of elements in an ordered pair is important and distinguishes it from an unordered pair, where the order of elements does not matter.

## 3. How are ordered pairs used in set theory?

In set theory, ordered pairs are often used to represent relationships between elements in different sets. For example, in the Cartesian product of two sets A and B, each element in A is paired with each element in B to form an ordered pair. This allows for the creation of new sets based on specific criteria and relationships between existing sets.

## 4. What is the difference between a set and an ordered pair?

A set is a collection of elements, while an ordered pair is a specific type of set that contains exactly two elements. Sets are often used to represent a group of objects or concepts, while ordered pairs are used to show a relationship or connection between two elements.

## 5. How is set theory applied in other fields of science?

Set theory has many applications in various fields of science, including computer science, physics, and biology. In computer science, it is used to analyze and manipulate data structures, while in physics it is used to study the relationships between mathematical models and physical phenomena. In biology, set theory is used to model genetic inheritance and population dynamics.

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