Question regarding algebra of sets

In summary, the algebra of sets is a mathematical system used to study the properties and relationships between sets. The basic operations in this algebra include union, intersection, and complement. It is commonly used in fields such as statistics, computer science, and economics to analyze and manipulate data sets, create Venn diagrams, and solve problems related to probability and logic. The main difference between the algebra of sets and arithmetic is that the former deals with sets while the latter deals with numbers. Some common mistakes when working with the algebra of sets include forgetting to include all elements in a set, incorrectly applying the operations, and confusing the order of operations. It is important to carefully think through each step and double-check all calculations to avoid mistakes.
  • #1
ssayani87
10
0

Homework Statement



I'm working on a proof for Intro. to Algebra, and the problem deals with proving that the operation of Symmetric Difference imposes group structure on the power set 2^x (set of all subsets) of a given set X. There is a part of my proof that I'd like to get some advice on however...

I've worked out a proof, and I can see it relies heavily on this equation:

Let A, B be sets.

i. A - B = (AuB) - B

Homework Equations



The Attempt at a Solution



My proof goes like this: (=>)Let x in A - B. Then, x is in A, implying x is in AuB. Since x cannot be in B, then it also falls in AuB - B.
(<=): Let x in AuB - B. Then x cannot be in B, and must therefore be in AuB. x not in B but still in AuB - B implies that x must be in A and not in B, which means it also falls in A - B.

Did I miss anything/ any mistakes?
 
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  • #2
ssayani87 said:

Homework Statement



I'm working on a proof for Intro. to Algebra, and the problem deals with proving that the operation of Symmetric Difference imposes group structure on the power set 2^x (set of all subsets) of a given set X. There is a part of my proof that I'd like to get some advice on however...

I've worked out a proof, and I can see it relies heavily on this equation:

Let A, B be sets.

i. A - B = (AuB) - B

Homework Equations



The Attempt at a Solution



My proof goes like this: (=>)Let x in A - B. Then, x is in A, implying x is in AuB. Since x cannot be in B, then it also falls in AuB - B.
(<=): Let x in AuB - B. Then x cannot be in B, and must therefore be in AuB. x not in B but still in AuB - B implies that x must be in A and not in B, which means it also falls in A - B.

Did I miss anything/ any mistakes?

Sounds fine to me.
 

What is the algebra of sets?

The algebra of sets is a mathematical system used to study the properties and relationships between sets. It involves combining sets using operations such as union, intersection, and complement.

What are the basic operations in the algebra of sets?

The basic operations in the algebra of sets are union, intersection, and complement. Union is the combination of all elements in two or more sets. Intersection is the common elements between two or more sets. Complement is the elements that are not in a specific set.

How is the algebra of sets used in real-life applications?

The algebra of sets is used in various fields such as statistics, computer science, and economics. It is used to analyze and manipulate data sets, create Venn diagrams, and solve problems related to probability and logic.

What is the difference between the algebra of sets and arithmetic?

The algebra of sets deals with sets, while arithmetic deals with numbers. In the algebra of sets, the operations are based on the elements within the sets, whereas in arithmetic, the operations are based on numerical values.

What are some common mistakes when working with the algebra of sets?

Some common mistakes when working with the algebra of sets include forgetting to include all elements in a set, incorrectly applying the operations, and confusing the order of operations. It is important to carefully think through each step and double-check all calculations to avoid mistakes.

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