https://r2.erweima.ai/mathbot/image/1726430386309zbpc57eo.jpg

Let's break down the given problem:

### Universal Set (ξ):
\[
\xi = \{ x : x \text{ is a positive number less than 13} \}
\]
This means:
\[
\xi = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 \}
\]

### Set \( G \) (Factors of 8):
\[
G = \{ x : x \text{ is a factor of 8} \}
\]
The factors of 8 are:
\[
G = \{ 1, 2, 4, 8 \}
\]

### Set \( H \) (Numbers greater than 9):
\[
H = \{ x : x \text{ is greater than 9} \}
\]
Thus:
\[
H = \{ 10, 11, 12 \}
\]

### Part (a): Is \( G \cap H' \) an empty set?

We need to find the intersection of \( G \) and \( H' \), where \( H' \) is the complement of \( H \) in \( \xi \). The complement \( H' \) consists of all elements in \( \xi \) that are *not* in \( H \), so:
\[
H' = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}
\]

Now, we find the intersection \( G \cap H' \):
\[
G \cap H' = \{ 1, 2, 4, 8 \} \cap \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}
\]
The common elements are:
\[
G \cap H' = \{ 1, 2, 4, 8 \}
\]
This is not an empty set.

### Part (b): Write down a set \( T \) which satisfies \( G' \cap T = \emptyset \).

Here, \( G' \) is the complement of \( G \) in \( \xi \), meaning all elements in \( \xi \) that are *not* in \( G \):
\[
G' = \{ 3, 5, 6, 7, 9, 10, 11, 12 \}
\]

For \( G' \cap T = \emptyset \), the set \( T \) must not contain any elements of \( G' \). Hence, \( T \) could be any subset of \( G \). For example:
\[
T = \{ 1, 2 \}
\]
This satisfies the condition since:
\[
G' \cap T = \emptyset
\]

Would you like further details on any part of the solution?

### Follow-up questions:
1. What is the complement of a set in set theory?
2. How do you find the intersection of two sets?
3. What is the significance of the universal set in problems like this?
4. How would you describe the complement \( H' \) in words?
5. Can you give another example of a set \( T \) that satisfies \( G' \cap T = \emptyset \)?

### Tip:
In set theory, the complement of a set \( A \) refers to all elements in the universal set that are *not* in \( A \). Always ensure you're working within the universal set when finding complements.

The universal set ξ, and sets G and H are defined below:
ξ = {x : x is a positive number less than 13}
G = {x : x is a factor of 8}
H = {x : x is greater than 9}

(a) Is G ∩ H' an empty set? Show your working.
(b) Write down a set, T, which satisfies G' ∩ T = ∅.

This set theory problem explores the intersection and complement of sets G and H within a universal set. Learn how to solve for G ∩ H' and find a set T such that G' ∩ T is an empty set. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation