Axioms of Set Theory .... and the Union of Two Sets ....

In summary, Hrbacek and Jech set up an axiomatic systems (which they do NOT call ZFC ... but it seems to mirror ZFC) that require the axiom of union and the axiom of pair to define the union of two sets.
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I am reading "Introduction to Set Theory" (Third Edition, Revised and Expanded) by Karel Hrbacek and Thomas Jech (H&J) ... ...

I am currently focused on Chapter 1: Sets and, in particular on Section 3: The Axioms where Hrbacek and Jech set up an axiomatic systems (which they do NOT call ZFC ... but it seems to mirror ZFC ... )

I need some help with Example 3.8 (c) which effectively defines the union of two sets ... . In particular I need some help in fully understanding how this definition requires H&J's Axiom of Pair and Axiom of Union in order to justify the definition ...Example 3.8, the notes following it and the Axiom of Pair and the Axiom of Union read as follows:https://www.physicsforums.com/attachments/7569
https://www.physicsforums.com/attachments/7570In some remarks following Example 3.8 (c) we read the following ... ...

" ... ... The Axiom of Pair and the Axiom of Union are necessary to define union of two sets ... ... "I can see how the Axiom of Union is necessary for the definition of the union of two sets ... BUT ... how exactly is the Union of Pair necessary? Indeed how is it implicated in this definition ... ...

*** NOTE *** The way I see it ... Example 3.8 (c) seems to 'work' perfectly if we assume that \(\displaystyle S = \{ M, N \}\) and 'apply' the Axiom of Union ... so ... nothing else seems to be needed ... except, of course, as Hrbacek and Jech note, the Axiom of Extensionality to guarantee that the union is unique ...Help will be appreciated ...

Peter====================================================================================
So that readers of the above post have access to Hrbacek and Jech's axiom system I am providing the relevant text ... as follows:
https://www.physicsforums.com/attachments/7571
https://www.physicsforums.com/attachments/7572
https://www.physicsforums.com/attachments/7573Hope access to the above text helps ...

Peter
 
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I need some help with Example 3.8 (c) which effectively defines the union of two sets ... . In particular I need some help in fully understanding how this definition requires H&J's Axiom of Pair and Axiom of Union in order to justify the definition ...
Hi Peter,

The way I see it, the axiom of union allows you to define the union of a set of sets.

To define $A\cup B$ using that axiom, you need a set whose elements are $A$ and $B$, and this requires the axiom of pairs.
 
  • #3
castor28 said:
Hi Peter,

The way I see it, the axiom of union allows you to define the union of a set of sets.

To define $A\cup B$ using that axiom, you need a set whose elements are $A$ and $B$, and this requires the axiom of pairs.
Yes ... what you suggest seems correct to me ...

Thanks for the help ... appreciate your assistance ...

Peter
 

What are the Axioms of Set Theory?

The Axioms of Set Theory are a set of fundamental principles that establish the rules for constructing and manipulating sets. These axioms form the foundation of modern mathematics and are used to define the concept of a set, as well as operations such as union, intersection, and complement.

What is the Union of Two Sets?

The union of two sets A and B is a set that contains all the elements that are in either A or B, or both. It is denoted by A ∪ B, and can also be thought of as the combination of the two sets. For example, if A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}.

What are the properties of the Union of Two Sets?

The union of two sets has several properties, including commutativity, associativity, and idempotence. Commutativity means that the order of the sets does not matter, so A ∪ B = B ∪ A. Associativity means that the grouping of sets does not change the result, so (A ∪ B) ∪ C = A ∪ (B ∪ C). Idempotence means that the union of a set with itself is the same set, so A ∪ A = A.

How is the Union of Two Sets represented in Venn diagrams?

In Venn diagrams, the union of two sets is represented by overlapping circles. The area where the circles overlap represents the elements that are in both sets, while the non-overlapping areas represent the elements that are unique to each set. For example, in a Venn diagram of A ∪ B, the overlapping area would contain all the elements in A and B, while the non-overlapping areas would contain the elements that are only in A or only in B.

What is the difference between Union and Intersection of Two Sets?

The union of two sets combines all the elements from both sets, while the intersection of two sets only includes the elements that are in both sets. In other words, the union is inclusive, while the intersection is exclusive. For example, if A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5} and A ∩ B = {3}.

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