Correctly order the following cardinalities: |R|, |N|, |Z|, |{u : u ∈ N}|
Understand the Problem
The question is asking to order the cardinalities of different sets: the set of real numbers (ℝ), the set of natural numbers (ℕ), the set of integers (ℤ), and a set defined by a specific property of natural numbers. This requires understanding the concept of cardinality in set theory.
Answer
$|\mathbb{N}| = |\mathbb{Z}| = |\{u : u \in \mathbb{N}\}| < |\mathbb{R}|$
Answer for screen readers
The ordered cardinalities from smallest to largest are: $|\mathbb{N}| = |\mathbb{Z}| = |{u : u \in \mathbb{N}}| < |\mathbb{R}|$
Steps to Solve
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Identify the cardinality of each set
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The set of natural numbers, denoted as $|\mathbb{N}|$, has a cardinality of $\aleph_0$, which represents countable infinity.
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The set of integers, denoted as $|\mathbb{Z}|$, also has a cardinality of $\aleph_0$ as it can be put into a one-to-one correspondence with the natural numbers, hence it is also countably infinite.
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The set of real numbers, denoted as $|\mathbb{R}|$, has a cardinality of $2^{\aleph_0}$, which represents uncountable infinity.
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The set defined by ${u : u \in \mathbb{N}}$ is basically the set of natural numbers, thus its cardinality is also $|\mathbb{N}| = \aleph_0$.
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Express the cardinalities symbolically
- Now we can summarize the cardinalities:
- $|\mathbb{R}| = 2^{\aleph_0}$
- $|\mathbb{N}| = \aleph_0$
- $|\mathbb{Z}| = \aleph_0$
- $|{u : u \in \mathbb{N}}| = \aleph_0$
- Now we can summarize the cardinalities:
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Order the cardinalities
- The cardinalities can be ordered based on their sizes:
- Since $\aleph_0 < 2^{\aleph_0}$, we have:
- $|\mathbb{N}| = \aleph_0$
- $|\mathbb{Z}| = \aleph_0$
- $|{u : u \in \mathbb{N}}| = \aleph_0$
- $|\mathbb{R}| = 2^{\aleph_0}$
- Since $\aleph_0 < 2^{\aleph_0}$, we have:
- The cardinalities can be ordered based on their sizes:
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Write the final order
- The correct order from smallest to largest is:
- $|\mathbb{N}| = |\mathbb{Z}| = |{u : u \in \mathbb{N}}| < |\mathbb{R}|$
- The correct order from smallest to largest is:
The ordered cardinalities from smallest to largest are: $|\mathbb{N}| = |\mathbb{Z}| = |{u : u \in \mathbb{N}}| < |\mathbb{R}|$
More Information
The cardinalities of $\mathbb{N}$ and $\mathbb{Z}$ are the same because both can be arranged in a sequence where each number corresponds to a unique natural number. The cardinality of $\mathbb{R}$ is larger, representing the continuum of real numbers.
Tips
Common mistakes include confusing countable and uncountable infinity or misunderstanding the notation used for cardinalities. To avoid these mistakes, it’s important to remember:
- Countable sets have cardinality $\aleph_0$.
- Uncountable sets, like the reals, have larger cardinalities.
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