Which of the following sets have the same cardinality? Select all that apply.
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![### Exercise: Identifying Sets with the Same Cardinality
**Question:**
Which of the following sets have the same cardinality? Select all that apply.
- □ \( (0,1) \)
- □ \( \mathbb{N} \)
- □ \( \mathbb{R} \)
- □ \( [0,1] \)
**Discussion:**
In set theory, two sets have the same cardinality if there is a one-to-one correspondence between the elements of the two sets. This concept helps us to understand different "sizes" of infinity, as displayed in various mathematical sets. In this context, you are asked to identify which sets among the given options have this property.
- **Option \( (0,1) \):** Represents the set of all real numbers between 0 and 1, not including 0 and 1.
- **Option \( \mathbb{N} \):** Represents the set of all natural numbers.
- **Option \( \mathbb{R} \):** Represents the set of all real numbers.
- **Option \( [0,1] \):** Represents the set of all real numbers between 0 and 1, including both 0 and 1.
Analyze the properties of these sets to determine their cardinality relationships.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b62c17d-5c05-48d6-ac2b-a3a7c31907f7%2Fecb19731-1a61-4522-87e9-df8e7a52bb17%2Fv4qxagn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Exercise: Identifying Sets with the Same Cardinality
**Question:**
Which of the following sets have the same cardinality? Select all that apply.
- □ \( (0,1) \)
- □ \( \mathbb{N} \)
- □ \( \mathbb{R} \)
- □ \( [0,1] \)
**Discussion:**
In set theory, two sets have the same cardinality if there is a one-to-one correspondence between the elements of the two sets. This concept helps us to understand different "sizes" of infinity, as displayed in various mathematical sets. In this context, you are asked to identify which sets among the given options have this property.
- **Option \( (0,1) \):** Represents the set of all real numbers between 0 and 1, not including 0 and 1.
- **Option \( \mathbb{N} \):** Represents the set of all natural numbers.
- **Option \( \mathbb{R} \):** Represents the set of all real numbers.
- **Option \( [0,1] \):** Represents the set of all real numbers between 0 and 1, including both 0 and 1.
Analyze the properties of these sets to determine their cardinality relationships.
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