Let's address question 8 from the tutorial sheet:

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Given:

• The universal set $E = \mathbb{R}$ (set of real numbers).
• The sets are defined as:
  • $X = \{ x \mid 3 < x < 6 \}$
  • $Y = \{ x \mid 5 \leq x \leq 7 \}$
  • $Z = \{ 0, 1, 2, 3 \}$

You are asked to find:

(i) $X \cap Y$
(ii) $X' \cap Z$
(iii) $(X \cap Y) \cap Z$
(iv) $X \cap (Y \cap Z)$

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Step-by-step Solutions:

(i) $X \cap Y$

We are given:

• $X = \{ x \mid 3 < x < 6 \}$
• $Y = \{ x \mid 5 \leq x \leq 7 \}$

The intersection $X \cap Y$ represents all elements that belong to both sets.

• $X = \{ 3 < x < 6 \}$, so $X = (3, 6)$.
• $Y = \{ 5 \leq x \leq 7 \}$, so $Y = [5, 7]$.

The intersection of these two sets is the overlap of the intervals:

• $X \cap Y = [5, 6)$.

(ii) $X' \cap Z$

$X'$ represents the complement of $X$ in $\mathbb{R}$, which means all real numbers except those between 3 and 6. So, $X'$ is:

• $X' = (-\infty, 3] \cup [6, \infty)$.

We are given:

• $Z = \{ 0, 1, 2, 3 \}$.

Now, the intersection $X' \cap Z$ represents all elements in $Z$ that are not in $X$, i.e., all elements of $Z$ that are less than or equal to 3 or greater than or equal to 6.

• $X' \cap Z = \{ 0, 1, 2, 3 \}$.

(iii) $(X \cap Y) \cap Z$

From part (i), we found that $X \cap Y = [5, 6)$. Now, we intersect this with $Z = \{ 0, 1, 2, 3 \}$:

• $(X \cap Y) \cap Z = \{ 0, 1, 2, 3 \} \cap [5, 6) = \emptyset$.

Thus, the result is an empty set:

• $(X \cap Y) \cap Z = \emptyset$.

(iv) $X \cap (Y \cap Z)$

First, we find the intersection of $Y$ and $Z$:

• $Y = [5, 7]$
• $Z = \{ 0, 1, 2, 3 \}$

The intersection $Y \cap Z$ is empty since there are no elements in $Z$ that are between 5 and 7.

Thus:

• $Y \cap Z = \emptyset$.

Now, intersect this result with $X$:

• $X \cap \emptyset = \emptyset$.

So:

• $X \cap (Y \cap Z) = \emptyset$.

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Summary of Results:

• $(i) \, X \cap Y = [5, 6)$
• $(ii) \, X' \cap Z = \{ 0, 1, 2, 3 \}$
• $(iii) \, (X \cap Y) \cap Z = \emptyset$
• $(iv) \, X \cap (Y \cap Z) = \emptyset$

Let me know if you'd like further clarification or additional help with any other problems! Here are some related questions:

1. How do you find the complement of a set in a specific universal set?
2. What is the difference between the intersection and union of sets?
3. How do you express the complement of an interval in set notation?
4. Can the intersection of two sets ever be empty? When does this happen?
5. How do you perform operations on sets when one set is defined by inequalities?

Tip: When working with set operations, always visualize the sets as intervals or number lines to better understand their relationships!