Set Cardinality and Counting Principles Quiz

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State the Cantor-Schröder-Bernstein Theorem.

If there exist injections $f: A \rightarrow B$ and $g: B \rightarrow A$, then there exists a bijection $h: A \rightarrow B$.

What is an uncountable set? Provide an example.

An uncountable set is a set that cannot be put in one-to-one correspondence with the natural numbers. An example is the set of real numbers.

Explain the Inclusion-Exclusion principle in combinatorics.

The Inclusion-Exclusion principle is a counting technique used to find the number of elements that belong to at least one of several sets, by subtracting the intersections of different sets.

Define equivalence relation on a set.

<p>An equivalence relation on a set is a binary relation that is reflexive, symmetric, and transitive.</p> Signup and view all the answers

How does the Pigeonhole Principle apply in combinatorics?

<p>The Pigeonhole Principle states that if there are more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon. In combinatorics, this principle is used to show the existence of repeated elements or outcomes when distributing objects into containers.</p> Signup and view all the answers