Proving |A U B| Given Disjoint A & B

  • Thread starter chocolatelover
  • Start date
In summary, the conversation discusses ways to prove the cardinality of the union of two disjoint sets, A and B. The first suggestion is to use a Venn diagram, while the second suggestion involves using indicator functions. The conversation also gives examples to help understand the concept.
  • #1
chocolatelover
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Hi everyone,

Could someone please show me how to prove this?

Homework Statement



Determine |A U B| in terms of |A| and |B| assuming that A and B are disjoint

Homework Equations





The Attempt at a Solution



I know that A U B must be finite because A and B are disjoint, but besides that I don't know how I would go about proving this.

Could someone please show me how to?

Thank you
 
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  • #2
There are some intuitive ways to answer this...
(i) The easiest way is to draw a Venn diagram and see what you think the answer might be
(ii) Secondly (and more formally) you could formulate the answer in terms of 'indicator functions'

[tex]i_X(x)= \begin{cases} 0 & \mbox{if }x \notin X \\ 1 & \mbox{if }x \in X [/tex]

Try the first part and then see if you can do the same via the second
 
  • #3
Thank you very much

Regards
 
  • #4
Just because A and B are disjoint, does not in any way imply that AuB is finite. Just count the elements (assuming both A and B are finite).
 
  • #5
Suppose A= {a}, B= {b}. What is AUB? What is |A|? What is |B|? What is |AUB|?

Suppose A= {a, b, c}, B= {u, v, w, x, y, z}. What is AUB? What is |A|? What is |B|? What is |AUB|?

Do those examples give you any ideas? When you have no idea how to do a general problem, look at simple examples.
 

1. How do you prove |A U B| given disjoint A & B?

To prove the cardinality of the union of two disjoint sets, you can use the formula |A U B| = |A| + |B|, where |A| and |B| represent the cardinality of sets A and B respectively. This formula works because disjoint sets have no elements in common, meaning that the total number of elements in the union is simply the sum of the elements in each set.

2. Can you provide an example to illustrate this concept?

Sure! Let's say we have two sets, A = {1, 2, 3} and B = {4, 5}. Since A and B are disjoint, their intersection is empty, and we can say that |A U B| = |A| + |B| = 3 + 2 = 5. This means that the union of A and B contains 5 elements, which are {1, 2, 3, 4, 5}.

3. What if A and B are not disjoint? How do you prove |A U B| in that case?

If A and B are not disjoint, then we need to consider the overlap between the two sets. In this case, the formula becomes |A U B| = |A| + |B| - |A ∩ B|, where |A ∩ B| represents the cardinality of the intersection of A and B. This accounts for the fact that the elements in the intersection are counted twice when we add the cardinalities of A and B.

4. Is it possible for |A U B| to be smaller than |A| or |B|?

No, it is not possible for the cardinality of the union to be smaller than the cardinality of either set. This is because even if A and B are disjoint, the union will always contain all the elements from both sets, meaning that it will have at least the same number of elements as the larger set.

5. Are there any other methods for proving the cardinality of a union?

Yes, there are other methods such as using Venn diagrams or using mathematical induction. However, the formula |A U B| = |A| + |B| (or |A U B| = |A| + |B| - |A ∩ B| for non-disjoint sets) is a simple and efficient way to prove the cardinality of a union, especially for larger sets.

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