Proving |A U B| Given Disjoint A & B

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SUMMARY

To prove |A U B| given that sets A and B are disjoint, one must recognize that the union of disjoint sets is simply the sum of their cardinalities. Specifically, |A U B| = |A| + |B| when both sets are finite. The discussion emphasizes using intuitive methods such as Venn diagrams and formal approaches like indicator functions to understand the concept. Examples provided illustrate the relationship between the sizes of the sets and their union.

PREREQUISITES
  • Understanding of set theory, specifically the concepts of unions and disjoint sets.
  • Familiarity with cardinality and how to calculate it for finite sets.
  • Basic knowledge of indicator functions and their applications in set theory.
  • Ability to interpret and create Venn diagrams for visual representation of sets.
NEXT STEPS
  • Study the properties of disjoint sets in set theory.
  • Learn about indicator functions and their role in mathematical proofs.
  • Explore Venn diagrams and their applications in visualizing set operations.
  • Practice calculating cardinalities of unions for various combinations of finite sets.
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Students studying set theory, mathematicians interested in combinatorial proofs, and educators teaching concepts of unions and cardinality in mathematics.

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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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'

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

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

Regards
 
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).
 
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.
 

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