Elementary Set Theory (Discrete)

In summary, the conversation discusses the concept of set differences and union, specifically in the context of A, B, and C being sets with A being a subset of B and B being a subset of C. It is noted that A/B and A/C are both empty sets, and the union of A and B is equivalent to just B. The concept is further explained in terms of set notation and examples are suggested for further understanding.
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rmiller70015
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Homework Statement


Suppose A⊂B⊂C. What is A/B, A/C, and A∪B

Homework Equations

The Attempt at a Solution


This isn't really a homework question, I am just trying to get some exposure to discrete math before I take it in the fall.

The set differences A/B and A/C are both empty sets and the 'or' set is B. I understand the last part, but I'm unsure of why A/B and A/C are empty sets. I understand it has something to do with x∈A but x∉B or C as part of the definition of a set difference. I just need someone to explain it to me.
 
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1. What is the definition of an element in set theory?

An element in set theory is any object or member that belongs to a set. It is represented by lowercase letters, such as x or y, and is denoted by the symbol "∈" which means "belongs to". For example, if we have the set A = {1, 2, 3}, then 2 is an element of A, represented as 2 ∈ A.

2. What is the difference between a set and a subset?

A set is a collection of distinct elements, while a subset is a set that contains only elements that are also in another set. In other words, all elements of a subset are also elements of the original set. For example, if A = {1, 2, 3} and B = {1, 2}, then B is a subset of A, denoted as B ⊆ A.

3. How do you represent the empty set in set theory?

The empty set, also known as the null set, is a set that contains no elements. It is denoted by the symbol "∅" or by using curly braces with no elements inside, {}. It is important to note that the empty set is a subset of every set, including itself.

4. What is the cardinality of a set?

The cardinality of a set is the number of elements it contains. It is denoted by the symbol "|" or "||" and is used to represent the size or count of a set. For example, if A = {1, 2, 3}, then the cardinality of A is 3, represented as |A| = 3.

5. How do you perform operations on sets in set theory?

In set theory, there are three main operations that can be performed on sets: union, intersection, and complement. Union (∪) combines all the elements from two sets, intersection (∩) finds the common elements between two sets, and complement (') finds all the elements that are not in a given set. These operations can be represented using Venn diagrams and can be used to solve many problems in mathematics and computer science.

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