What is the Definition of a Relation in Set Theory?

In summary, the exercise states that a set ##A## is a relation if and only if it is a subset of the Cartesian product of its domain and range. The forward direction is straightforward, but the other direction is confusing because it uses the dom and ran operations, which are typically defined for relations. It is clarified that these operations can be defined for arbitrary sets, such as in the example where ##A = \{1, (2,3)\}##. The definition of "relation" used in this exercise is equivalent to the commonly seen definition of the Cartesian product of a set's domain and range.
  • #1
Mr Davis 97
1,462
44
I have an exercise in my set theory book that states the following: Show that a set ##A## is a relation iff ##A \subseteq \operatorname{dom} A\times \operatorname{ran} B##.

This is an easy exercise, so I am not asking how to prove it. However, I am confused about one thing.

The forward direction is trivial, by the definition of the relation. However, the other direction confuses me, because it uses the dom and ran operations, which I thought were only defined for relations, when in fact we can't assume that ##A## is a relation. Is this saying that dom and ran are defined with respect to arbitrary sets? For example, if ##A = \{1, (2,3)\}##, then dom A = 2 and ran A = 3, where the 1 is just disregarded?
 
Physics news on Phys.org
  • #2
What definition of "relation" do you have? The thing you are asked to prove to be equivalent is what I am used to seeing as the definition.
I would assume that "dom" and "ran" were just inserted to help you understand what roles A and B have, respectively.
 

What is "Terminology for relations"?

"Terminology for relations" refers to the specific language or vocabulary used to describe and define relationships between objects or concepts. It is often used in fields such as mathematics, logic, and computer science.

Why is terminology for relations important?

Having a standardized and precise set of terms for describing relationships allows for clear communication and understanding among scientists and researchers. It also allows for consistency and accuracy in data analysis and interpretation.

What are some common terms used in terminology for relations?

Some common terms include "is a subset of," "implies," "equivalent to," and "is not related to." There are also specific terms used for different types of relationships, such as "parent-child," "sibling," and "ancestor-descendant."

How is terminology for relations related to logic and mathematics?

Terminology for relations is closely related to logic and mathematics because it involves the precise description and analysis of relationships between objects or concepts. In logic, terminology for relations is used to establish the validity of arguments and proofs. In mathematics, it is used to define and describe mathematical concepts and structures.

How does terminology for relations differ from other types of terminology?

Terminology for relations is unique in that it specifically focuses on the language used to describe relationships between objects or concepts. Other types of terminology may focus on different aspects, such as definitions, classifications, or characteristics of objects or concepts.

Similar threads

MHB The Recursion Theorem .... Searcoid, Theroem 1.3.24 .... ....
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
I Does the definition of cardinality assume distinguishability?
    • Set Theory, Logic, Probability, Statistics
Replies
9
Views
1K
I Thinking about equality of infinite sets
    • Set Theory, Logic, Probability, Statistics
Replies
5
Views
1K
I May I use set theory to define the number of solutions of polynomials?
    • Set Theory, Logic, Probability, Statistics
Replies
13
Views
2K
A Defining the membership relation in set theory?
    • Set Theory, Logic, Probability, Statistics
Replies
14
Views
4K
MHB Upper Bound of Sets and Sequences: Analyzing Logic
    • Set Theory, Logic, Probability, Statistics
Replies
3
Views
4K
MHB Show that ~ is an equivalence relation
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
MHB What are the properties of partial order relations according to J&W's book?
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
I Subset definition: universal quantifier over which set?
    • Set Theory, Logic, Probability, Statistics
Replies
33
Views
3K
MHB Sets so that the cartesian product is commutative
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top