Hard Time Understanding If A C B, then A U B = B

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In summary: To show A ∪ B = B is possible, you need to show that every element of A ∪ B is an element of B and that every element of B is an element of A ∪ B.
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Hard Time Understanding "If A C B, then A U B = B"

Homework Statement



Understand how "If A [tex]\supseteq[/tex] B, then A U B = B" is possible.


Homework Equations



None.


The Attempt at a Solution



Since A U B = B, it can be separated into two cases. That is,

1) A U B [tex]\subseteq[/tex] B
2) B [tex]\subseteq[/tex] A U B

For case (1), I let x [tex]\in[/tex] A U B. Thus, either x [tex]\in[/tex] A or x [tex]\in[/tex] B. If x [tex]\in[/tex] A, then x [tex]\in[/tex] B. This means that A U B [tex]\subseteq[/tex] B.

For case (2), I let x [tex]\in[/tex] B. This is where I got stuck... I know I am supposed to apply the assumption that A [tex]\supseteq[/tex] B, but I am starting to think that it is impossible.

Here is an attachment of why I think it is impossible. Here is the link to tinypic for those who are too afraid to download attachments: http://tinypic.com/view.php?pic=xszle&s=7.

Can anyone give me a tip on how to approach this problem?
 

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  • #2


number0 said:

Homework Statement



Understand how "If A [tex]\supseteq[/tex] B, then A U B = B" is possible.


Homework Equations



None.


The Attempt at a Solution



Since A U B = B, it can be separated into two cases. That is,

1) A U B [tex]\subseteq[/tex] B
2) B [tex]\subseteq[/tex] A U B

For case (1), I let x [tex]\in[/tex] A U B. Thus, either x [tex]\in[/tex] A or x [tex]\in[/tex] B. If x [tex]\in[/tex] A, then x [tex]\in[/tex] B. This means that A U B [tex]\subseteq[/tex] B.

For case (2), I let x [tex]\in[/tex] B. This is where I got stuck...

But if x ε B then isn't x ε A U B?
 
  • #3


number0 said:
Understand how "If A [tex]\supseteq[/tex] B, then A U B = B" is possible.

What you wrote holds only if A=B. Another scenario would be [itex] A\subseteq B \Rightarrow A\cup B = B [/itex]

which is trivial.
 
  • #4


number0 said:
Understand how "If A [tex]\supseteq[/tex] B, then A U B = B" is possible.
From the other stuff you wrote, I'm assuming this is a typo and you meant if A ⊂ B, then A ∪ B = B.

In your attachment, you interpreted B ⊂ A ∪ B to mean B is a subset of A and a subset of B. This isn't correct. Say C ⊂ A ∪ B. If x∈C, that only means x∈A or x∈B. It's possible for some elements of C to be only in A and others to be only in B so that C is a subset of neither A nor B. So B ⊂ A ∪ B doesn't tell you anything about whether B is a subset of A or a subset of B. It just says all the elements in B are also elements in the union of A and B.
 

1. What does "A C B" mean in the statement "If A C B, then A U B = B"?

In set theory, "A C B" means that A is a subset of B. This means that every element in A is also in B. This can also be read as "A is contained in B".

2. What does "A U B" mean in the statement "If A C B, then A U B = B"?

In set theory, "A U B" means the union of sets A and B, which is the set of all elements that are in either A, B, or both.

3. How is the statement "If A C B, then A U B = B" true?

Since A is a subset of B, every element in A is also in B. This means that when we take the union of A and B, we are essentially adding all the elements that are in A to the set B. Since all the elements in A are already in B, the union will not change B. Therefore, A U B = B.

4. Can you provide an example to illustrate the statement "If A C B, then A U B = B"?

Let's say A = {1, 2} and B = {1, 2, 3}. Since every element in A (1 and 2) is also in B, we can say that A is a subset of B, or A C B. When we take the union of A and B, we get {1, 2, 3}. Since all the elements in A are already in B, the union does not change B. Therefore, A U B = B.

5. How does understanding "A C B" help us understand the statement "If A C B, then A U B = B"?

Understanding "A C B" helps us understand the statement "If A C B, then A U B = B" by giving us the necessary context and definition of the term "subset". This helps us understand why the statement is true and how it can be applied to different examples in mathematics and science.

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