Part a) Set Theory Operations

i) $D \cup E$ (Union of $D$ and $E$)

The union of two sets $D$ and $E$ is the set of all elements that belong to either $D$ or $E$ (or both).

Given:

• $D = \{1, 3, 4, 6\}$
• $E = \{4, 5, 6\}$

Union $D \cup E$ is: $D \cup E = \{1, 3, 4, 5, 6\}$

ii) $D'$ (Complement of $D$)

The complement of $D$, denoted $D'$, includes all the elements from the universal set that are not in $D$.
Given the universal set $U = \{1, 2, 3, 4, 5, 6\}$ and $D = \{1, 3, 4, 6\}$, we find:

$D' = U - D = \{2, 5\}$

iii) Show that $D - E \neq E - D$

To show this, we calculate the set difference $D - E$ and $E - D$.

• $D - E$ contains the elements of $D$ that are not in $E$: $D - E = \{1, 3, 4, 6\} - \{4, 5, 6\} = \{1, 3\}$

• $E - D$ contains the elements of $E$ that are not in $D$: $E - D = \{4, 5, 6\} - \{1, 3, 4, 6\} = \{5\}$

Clearly, $D - E = \{1, 3\} \neq \{5\} = E - D$.

Thus, $D - E \neq E - D$.

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Part b) Inequality in Interval Notation and Number Line

Express $-4 \leq x < 10$ in interval notation

The inequality $-4 \leq x < 10$ represents all values of $x$ from $-4$ to 10, including $-4$ but excluding 10. In interval notation:

$[-4, 10)$

Show on the number line

To represent this on the number line:

• Use a closed circle at $-4$ to indicate that $-4$ is included.
• Use an open circle at $10$ to indicate that $10$ is not included.
• Draw a line between $-4$ and $10$, showing all values between them.

Let me know if you'd like me to go into more detail or have any questions!

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Follow-up Questions:

1. What is the intersection of two sets, and how does it differ from the union?
2. How can the complement of a set be used in Venn diagrams?
3. What is the difference between a closed and open interval in interval notation?
4. What is the general formula for the symmetric difference of two sets?
5. How would you apply set difference in a real-world scenario?

Tip: When working with set differences, always remember to subtract the elements of one set from another, and be sure to check if an element belongs to both sets or only one.