Unraveling the Explanation of A \bigcup B, A \bigcap B, and A X B

[XodoX]

I don't understand it.

A has n elements, and B has m elements. Give the exact maximum/minimum of

1) A $\bigcup$ B

2) A $\bigcap$ B

3) A X BI don't understand the solution to this..

1) If A and B are a disjunction ( A$\bigcap$ B = ∅), then the max of A $\bigcup$ B is:

A $\bigcap$ B = ∅ -> |A$\bigcup$B| = m + nIf A is a subset of B (A$\subseteq$B) or B a subset of A (B $\subseteq$ A),
then the min of A and B is:

A$\subseteq$B -> |A$\bigcup$B| = |B| = m

B$\subseteq$A -> |A$\bigcup$B| = |A| = nSo you're basically saying the min here is m and n. I understand that. I just don't get the explanation of it. I have to show why it's the min.

Therefore, the max of A $\bigcup$ B is:

max(n,m) $\leq$ |A$\bigcup$B| $\leq$ n+m

Don't get this one. In words: The max is no greater than n+m. But it says it's less or equal to A and B. So you're already assuming A and B is the max?

2 and 3 have the same confusing explanations.

 

[chiro]

Hey XodoX.

Hint: |A OR B| = |A| + |B| - |A AND B|

 

[HallsofIvy]

Suppose A contains n elements and B contains m elements with m< n.

Now consider two extreme cases:
1) A and B are completely disjoint (they have no elements in common)
Then $|A\cup B|= m+n$ and $|A\cap B|= 0$.
2) B is a subset of A
Then $|A\cup B|= n$ and $|A\cap B|= m$

 

[XodoX]

Hey XodoX.

Hint: |A OR B| = |A| + |B| - |A AND B|

Yes, I know, but I don't what you're referring to.

Suppose A contains n elements and B contains m elements with m< n.

Now consider two extreme cases:
1) A and B are completely disjoint (they have no elements in common)
Then $|A\cup B|= m+n$ and $|A\cap B|= 0$.
2) B is a subset of A
Then $|A\cup B|= n$ and $|A\cap B|= m$

So the empty set mean nothing in common. I have A and B, so the max is A+B. Like having two separate balls.
But the A or B... if they are disjoint, it says it's 0. That would mean A and B have also a min that is 0, but there's only a max. Shouldn't A and B and A or B both have max and min ?

I don't understand your 2). A is n and B is m. So B is a subset of A means A "swallows" B and, therefore, it's A, or n. And A is a subset of B means B "swallows" A and the result is B, or m.
Is that kind of like this?BTW. To solve this, I always have to show disjoint and subset? And "and" has always just a max and "or" only a min ?

 

[verty]

Read online about Venn diagrams, this may help.

 

[HallsofIvy]

Yes, I know, but I don't what you're referring to.



So the empty set mean nothing in common. I have A and B, so the max is A+B.

You are using "A" and "B" to mean both the sets and the cardinality of the sets. Don't do that!
Yes, the intersection of two sets is empty if and only if they have "nothing in common".
Use precise language.

Like having two separate balls.
But the A or B... if they are disjoint, it says it's 0.

What does 'it' refer to and why would it say anything?

That would mean A and B have also a min that is 0, but there's only a max.

Now you are talking nonsense. A and B are general sets, NOT necessarily sets of numbers and do not necessarily have a "max" or "min". If, by "A" and "B" you mean their cardinalities (again, bad notation) they are fixed sets with fixed cardinality so again it is nonsense to talk about "max" and "min".

Shouldn't A and B and A or B both have max and min ?

I don't understand your 2). A is n and B is m. So B is a subset of A means A "swallows" B and, therefore, it's A, or n. And A is a subset of B means B "swallows" A and the result is B, or m.
Is that kind of like this?


BTW. To solve this, I always have to show disjoint and subset? And "and" has always just a max and "or" only a min ?