Example 2.5: Write Down The Following Sets As A List Of Elements:1. ${x \in \mathbb{R} \mid X \in \mathbb{Z} \text{ And } 0 \leq X \leq 5}$2. { X ∈ R ∣ 3 X ∈ Z And 0 \textless X ≤ 2 } \{x \in \mathbb{R} \mid 3x \in \mathbb{Z} \text{ And } 0 \ \textless \ X \leq 2\} { X ∈ R ∣ 3 X ∈ Z And 0 \textless X ≤ 2 }

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Understanding the Problem

In this example, we are given two sets defined using set-builder notation. Our goal is to write down each set as a list of elements. This involves understanding the conditions specified in the set-builder notation and identifying the elements that satisfy those conditions.

Set 1: {x∈R∣x∈Z and 0≤x≤5}\{x \in \mathbb{R} \mid x \in \mathbb{Z} \text{ and } 0 \leq x \leq 5\}

The first set is defined as follows:

  • The set contains real numbers (x∈Rx \in \mathbb{R}).
  • The set also contains integers (x∈Zx \in \mathbb{Z}).
  • The integer xx must satisfy the condition 0≤x≤50 \leq x \leq 5.

To write down this set as a list of elements, we need to find all the integers between 0 and 5 inclusive.

Set 1: {0, 1, 2, 3, 4, 5}

Set 2: {x∈R∣3x∈Z and 0 \textless x≤2}\{x \in \mathbb{R} \mid 3x \in \mathbb{Z} \text{ and } 0 \ \textless \ x \leq 2\}

The second set is defined as follows:

  • The set contains real numbers (x∈Rx \in \mathbb{R}).
  • The set also contains real numbers such that 3x3x is an integer (3x∈Z3x \in \mathbb{Z}).
  • The real number xx must satisfy the condition 0 \textless x≤20 \ \textless \ x \leq 2.

To write down this set as a list of elements, we need to find all the real numbers between 0 and 2 inclusive such that 3x3x is an integer.

Set 2: {1/3, 2/3}

Explanation

In the second set, we need to find all the real numbers between 0 and 2 inclusive such that 3x3x is an integer. This means that 3x3x must be a multiple of 3. To find the values of xx, we can divide the multiples of 3 by 3.

The multiples of 3 between 0 and 6 inclusive are 3 and 6. Dividing these multiples by 3 gives us the values of xx as 1 and 2. However, we need to find the values of xx between 0 and 2 inclusive. Therefore, we only consider the value x=1/3x = 1/3 and x=2/3x = 2/3.

Conclusion

In this example, we have written down two sets defined using set-builder notation as a list of elements. We have identified the elements that satisfy the conditions specified in the set-builder notation and listed them out. This involves understanding the conditions specified in the set-builder notation and identifying the elements that satisfy those conditions.

Key Takeaways

  • Set-builder notation is a way of defining a set using a property or condition that the elements of the set must satisfy.
  • To write down a set defined using set-builder notation as a list of elements, we need to identify the elements that satisfy the conditions specified in the set-builder notation.
  • The conditions specified in the set-builder notation must be satisfied by all the elements in the set.

Further Reading

For more information on set-builder notation and writing sets as a list of elements, please refer to the following resources:

Practice Problems

Try the following practice problems to test your understanding of writing sets as a list of elements:

  • Write down the set {x∈R∣x∈Z and −2≤x≤2}\{x \in \mathbb{R} \mid x \in \mathbb{Z} \text{ and } -2 \leq x \leq 2\} as a list of elements.
  • Write down the set {x∈R∣2x∈Z and 0 \textless x≤3}\{x \in \mathbb{R} \mid 2x \in \mathbb{Z} \text{ and } 0 \ \textless \ x \leq 3\} as a list of elements.
    Q&A: Writing Sets as a List of Elements =============================================

Frequently Asked Questions

In this article, we will answer some frequently asked questions about writing sets as a list of elements.

Q: What is set-builder notation?

A: Set-builder notation is a way of defining a set using a property or condition that the elements of the set must satisfy. It is a shorthand way of writing a set using a mathematical expression.

Q: How do I write a set defined using set-builder notation as a list of elements?

A: To write a set defined using set-builder notation as a list of elements, you need to identify the elements that satisfy the conditions specified in the set-builder notation. This involves understanding the conditions specified in the set-builder notation and identifying the elements that satisfy those conditions.

Q: What are some common mistakes to avoid when writing sets as a list of elements?

A: Some common mistakes to avoid when writing sets as a list of elements include:

  • Not understanding the conditions specified in the set-builder notation.
  • Not identifying all the elements that satisfy the conditions specified in the set-builder notation.
  • Including elements that do not satisfy the conditions specified in the set-builder notation.

Q: How do I determine if an element is in a set defined using set-builder notation?

A: To determine if an element is in a set defined using set-builder notation, you need to check if the element satisfies the conditions specified in the set-builder notation. This involves substituting the element into the mathematical expression and checking if it is true.

Q: Can I use set-builder notation to define a set that contains only one element?

A: Yes, you can use set-builder notation to define a set that contains only one element. For example, the set {x∈R∣x=5}\{x \in \mathbb{R} \mid x = 5\} contains only the element 5.

Q: Can I use set-builder notation to define a set that contains no elements?

A: Yes, you can use set-builder notation to define a set that contains no elements. For example, the set {x∈R∣x≠5}\{x \in \mathbb{R} \mid x \neq 5\} contains no elements.

Q: How do I write a set that contains all real numbers as a list of elements?

A: You cannot write a set that contains all real numbers as a list of elements. This is because there are infinitely many real numbers, and it is not possible to list them all out.

Q: How do I write a set that contains all integers as a list of elements?

A: You cannot write a set that contains all integers as a list of elements. This is because there are infinitely many integers, and it is not possible to list them all out.

Q: Can I use set-builder notation to define a set that contains all elements of another set?

A: Yes, you can use set-builder notation to define a set that contains all elements of another set. For example, the set {x∈R∣x∈Z}\{x \in \mathbb{R} \mid x \in \mathbb{Z}\} contains all integers.

Q: Can I use set-builder notation to define a set that contains only elements of a certain type?

A: Yes, you can use set-builder notation to define a set that contains only elements of a certain type. For example, the set {x∈R∣x is an integer}\{x \in \mathbb{R} \mid x \text{ is an integer}\} contains only integers.

Q: Can I use set-builder notation to define a set that contains elements that satisfy multiple conditions?

A: Yes, you can use set-builder notation to define a set that contains elements that satisfy multiple conditions. For example, the set {x∈R∣x∈Z and 0≤x≤5}\{x \in \mathbb{R} \mid x \in \mathbb{Z} \text{ and } 0 \leq x \leq 5\} contains elements that are integers and satisfy the condition 0≤x≤50 \leq x \leq 5.

Q: Can I use set-builder notation to define a set that contains elements that satisfy a condition that involves a function?

A: Yes, you can use set-builder notation to define a set that contains elements that satisfy a condition that involves a function. For example, the set {x∈R∣sin⁡(x)∈Z}\{x \in \mathbb{R} \mid \sin(x) \in \mathbb{Z}\} contains elements that satisfy the condition sin⁡(x)∈Z\sin(x) \in \mathbb{Z}.

Q: Can I use set-builder notation to define a set that contains elements that satisfy a condition that involves a relation?

A: Yes, you can use set-builder notation to define a set that contains elements that satisfy a condition that involves a relation. For example, the set {x∈R∣x is related to 5}\{x \in \mathbb{R} \mid x \text{ is related to } 5\} contains elements that are related to 5.

Q: Can I use set-builder notation to define a set that contains elements that satisfy a condition that involves a property?

A: Yes, you can use set-builder notation to define a set that contains elements that satisfy a condition that involves a property. For example, the set {x∈R∣x is a prime number}\{x \in \mathbb{R} \mid x \text{ is a prime number}\} contains elements that are prime numbers.

Q: Can I use set-builder notation to define a set that contains elements that satisfy a condition that involves a set operation?

A: Yes, you can use set-builder notation to define a set that contains elements that satisfy a condition that involves a set operation. For example, the set {x∈R∣x∈Z and x∈N}\{x \in \mathbb{R} \mid x \in \mathbb{Z} \text{ and } x \in \mathbb{N}\} contains elements that are integers and natural numbers.

Q: Can I use set-builder notation to define a set that contains elements that satisfy a condition that involves a logical operator?

A: Yes, you can use set-builder notation to define a set that contains elements that satisfy a condition that involves a logical operator. For example, the set {x∈R∣x∈Z or x∈N}\{x \in \mathbb{R} \mid x \in \mathbb{Z} \text{ or } x \in \mathbb{N}\} contains elements that are integers or natural numbers.

Q: Can I use set-builder notation to define a set that contains elements that satisfy a condition that involves a quantifier?

A: Yes, you can use set-builder notation to define a set that contains elements that satisfy a condition that involves a quantifier. For example, the set {x∈R∣∀y∈R,y≤x}\{x \in \mathbb{R} \mid \forall y \in \mathbb{R}, y \leq x\} contains elements that satisfy the condition ∀y∈R,y≤x\forall y \in \mathbb{R}, y \leq x.

Q: Can I use set-builder notation to define a set that contains elements that satisfy a condition that involves a universal quantifier?

A: Yes, you can use set-builder notation to define a set that contains elements that satisfy a condition that involves a universal quantifier. For example, the set {x∈R∣∀y∈R,y≤x}\{x \in \mathbb{R} \mid \forall y \in \mathbb{R}, y \leq x\} contains elements that satisfy the condition ∀y∈R,y≤x\forall y \in \mathbb{R}, y \leq x.

Q: Can I use set-builder notation to define a set that contains elements that satisfy a condition that involves an existential quantifier?

A: Yes, you can use set-builder notation to define a set that contains elements that satisfy a condition that involves an existential quantifier. For example, the set {x∈R∣∃y∈R,y≤x}\{x \in \mathbb{R} \mid \exists y \in \mathbb{R}, y \leq x\} contains elements that satisfy the condition ∃y∈R,y≤x\exists y \in \mathbb{R}, y \leq x.

Q: Can I use set-builder notation to define a set that contains elements that satisfy a condition that involves a negation operator?

A: Yes, you can use set-builder notation to define a set that contains elements that satisfy a condition that involves a negation operator. For example, the set {x∈R∣¬(x∈Z)}\{x \in \mathbb{R} \mid \neg (x \in \mathbb{Z})\} contains elements that do not satisfy the condition x∈Zx \in \mathbb{Z}.

Q: Can I use set-builder notation to define a set that contains elements that satisfy a condition that involves a conjunction operator?

A: Yes, you can use set-builder notation to define a set that contains elements that satisfy a condition that involves a conjunction operator. For example, the set {x∈R∣x∈Z and x∈N}\{x \in \mathbb{R} \mid x \in \mathbb{Z} \text{ and } x \in \mathbb{N}\} contains elements that are integers and natural numbers.

Q: Can I use set-builder notation to define a set that contains elements that satisfy a condition that involves a disjunction operator?

A: Yes, you can use set-builder notation to define a set that contains elements that satisfy a condition that involves a disjunction operator. For example, the set {x∈R∣x∈Z or x∈N}\{x \in \mathbb{R} \mid x \in \mathbb{Z} \text{ or } x \in \mathbb{N}\} contains elements that are integers or natural numbers.

Q: Can I use set-builder notation to define a set that contains elements that satisfy a condition that involves an implication operator?

A: Yes, you can use set-builder notation to define a set that contains elements that satisfy a condition that involves an implication operator. For example, the set ${x \in \mathbb{R} \