Set Theory Problems: S1 U S2 = (S1' ∩ S2')' and S1 U S2 - (S1 ∩ S2') = S2

In summary: Here is a start for part 2:Let x ∈ S1 U S2 - (S1 ∩ S2')Then x ∈ S1 or x ∈ S2 and x ∉ S1 ∩ S2'Consider the two cases separately:Case 1: x ∈ S1Then x ∈ S1 U S2 and x ∉ S1 ∩ S2'So x ∈ S2 (why?)Case 2: x ∈ S2Then x ∈ S1 U S2 and x ∉ S1 ∩ S2'So x ∈ S2 (why?)Therefore, in both cases, x ∈ S2. Thus, S1 U S2 - (S1
  • #1
sbc824
5
0

Homework Statement



show S1 U S2 = (S1' ∩ S2')'

The Attempt at a Solution



I'm pretty sure I have this right or I'm close

Let x ∈ S1 U S2
x ∈ S1 or x ∈ S2
Since x ∈ S1 or S2, then x ∉ S1' and S2'
If x ∉ S1' and S2', then x ∈ (S1' and S2')'
Therefore, S1 U S2 = (S1' ∩ S2')'

Homework Statement



show S1 U S2 - (S1 ∩ S2') = S2

The Attempt at a Solution



I have not attempted this as I'm not sure how to start this one...any help would be appreciated.
 
Last edited:
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  • #2
sbc824 said:

Homework Statement



show S1 U S2 = (S1' ∩ S2')'

The Attempt at a Solution



I'm pretty sure I have this right or I'm close

Let x ∈ S1 U S2
x ∈ S1 or x ∈ S2
Since x ∈ S1 or x ∈ S2, then x ∉ S1' and x ∉ S2' This is not correct.

If x ∈ S1, then it does not need to be in S2. If it's not in S2, then x ∈ S2'.


If x ∉ S1' and x ∉ S2', then x ∈ (S1' and S2')'
Therefore, S1 U S2 = (S1' ∩ S2')'

Homework Statement



show S1 U S2 - (S1 ∩ S2') = S2

The Attempt at a Solution



I have not attempted this as I'm not sure how to start this one...any help would be appreciated.
You don't have the first part right.
 
  • #3
SammyS said:
You don't have the first part right.

wow silly mistake thanks...any starting hints for 2? I can easily visualize it with a diagram...but I'm rusty with set notation.
 
  • #4
sbc824 said:
wow silly mistake thanks...any starting hints for 2? I can easily visualize it with a diagram...but I'm rusty with set notation.
Another problem with your solution to part 1 is that you have only shown that S1 U S2 ⊆ (S1' ∩ S2')' (that is, if you have truly corrected your proof). To show equality, you also need to show that S1 U S2 ⊇ (S1' ∩ S2')' .
 

1. What is set theory?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects. It is the foundation of modern mathematics and is used in many fields, including computer science, statistics, and physics.

2. What are set theory problems?

Set theory problems are mathematical questions or challenges that require the application of set theory concepts and principles to solve. These problems can range from basic set operations to more complex concepts like cardinality and Venn diagrams.

3. How do I approach solving set theory problems?

The first step in solving set theory problems is to understand the given problem and identify the relevant sets and their elements. Then, use the appropriate set operations and principles to find the solution. It is also helpful to draw Venn diagrams to visualize the problem and its solution.

4. What are some common set theory problems?

Some common set theory problems include finding the union, intersection, and complement of sets, determining the cardinality of sets, and solving problems using Venn diagrams. These problems can range from simple to complex and are often used in various mathematical and scientific fields.

5. Why is set theory important in science?

Set theory is important in science because it provides a framework for understanding and organizing the relationships between different elements or objects. It allows scientists to categorize and analyze data, make predictions, and develop theories. It is also used in various scientific fields, such as biology, physics, and computer science, to model and solve complex problems.

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