Let R, S and T be sets defined as follows.
R = {2,3, 6, 7}
S = {(2), {2,3, 4}, {2, 5, 6}, {6, 7) }
T = {x is an element of Z: x ≤ 3 or x ≥ 7}
Find the following.
(i) R -T
(ii) S - P(R)
(iii) P(R)  intersect P(T)
(iv) (R - {3,4,5,6}) x (Z - T)
(v) |(P(R) - S) x S|

Let's solve each of these step by step.

### Given Sets:
- \( R = \{2, 3, 6, 7\} \)
- \( S = \{(2), \{2, 3, 4\}, \{2, 5, 6\}, \{6, 7\}\} \)
- \( T = \{x \in \mathbb{Z} : x \leq 3 \text{ or } x \geq 7\} \)

### (i) \( R - T \)

The set \( T \) consists of all integers \( x \) such that \( x \leq 3 \) or \( x \geq 7 \). So:

\[
T = \{\dots, -2, -1, 0, 1, 2, 3, 7, 8, 9, \dots \}
\]

Now, we find \( R - T \), which means the elements in \( R \) that are not in \( T \):

\[
R - T = \{2, 3, 6, 7\} - \{\dots, -2, -1, 0, 1, 2, 3, 7, 8, 9, \dots \} = \{6\}
\]

So, \( R - T = \{6\} \).

### (ii) \( S - P(R) \)

First, let's find \( P(R) \), the power set of \( R \). The power set \( P(R) \) is the set of all subsets of \( R \):

\[
P(R) = \{\emptyset, \{2\}, \{3\}, \{6\}, \{7\}, \{2, 3\}, \{2, 6\}, \{2, 7\}, \{3, 6\}, \{3, 7\}, \{6, 7\}, \{2, 3, 6\}, \{2, 3, 7\}, \{2, 6, 7\}, \{3, 6, 7\}, \{2, 3, 6, 7\}\}
\]

Now, we find \( S - P(R) \):

\[
S - P(R) = \{(2), \{2, 3, 4\}, \{2, 5, 6\}, \{6, 7\}\} - P(R)
\]

Since \( (2) \), \{2, 3, 4\}, and \{2, 5, 6\} are not subsets of \( R \), they will remain in \( S - P(R) \):

\[
S - P(R) = \{(2), \{2, 3, 4\}, \{2, 5, 6\}\}
\]

### (iii) \( P(R) \cap P(T) \)

Let's determine \( P(T) \), the power set of \( T \). Since \( T = \{x \in \mathbb{Z} : x \leq 3 \text{ or } x \geq 7\} \) is an infinite set, we'll restrict our consideration to elements in \( P(T) \) that intersect with subsets of \( P(R) \).

Since \( R = \{2, 3, 6, 7\} \), \( P(T) \) includes subsets such as \( \{2\}, \{3\}, \{7\} \) but not \( \{6\} \) since 6 is not in \( T \).

So, the intersection \( P(R) \cap P(T) \) consists of the subsets of \( R \) that also belong to \( P(T) \):

\[
P(R) \cap P(T) = \{\emptyset, \{2\}, \{3\}, \{7\}, \{2, 3\}, \{2, 7\}, \{3, 7\}, \{2, 3, 7\}\}
\]

### (iv) \( (R - \{3, 4, 5, 6\}) \times (Z - T) \)

Let's find \( R - \{3, 4, 5, 6\} \):

\[
R - \{3, 4, 5, 6\} = \{2, 3, 6, 7\} - \{3, 4, 5, 6\} = \{2, 7\}
\]

Now, \( Z - T \) is the set of integers \( x \) such that \( 4 \leq x \leq 6 \):

\[
Z - T = \{4, 5, 6\}
\]

The Cartesian product \( (R - \{3, 4, 5, 6\}) \times (Z - T) \) is:

\[
\{(2, 4), (2, 5), (2, 6), (7, 4), (7, 5), (7, 6)\}
\]

### (v) \( |(P(R) - S) \times S| \)

We already know that:

\[
P(R) = \{\emptyset, \{2\}, \{3\}, \{6\}, \{7\}, \{2, 3\}, \{2, 6\}, \{2, 7\}, \{3, 6\}, \{3, 7\}, \{6, 7\}, \{2, 3, 6\}, \{2, 3, 7\}, \{2, 6, 7\}, \{3, 6, 7\}, \{2, 3, 6, 7\}\}
\]

\( P(R) - S \) would remove any elements of \( P(R) \) that are in \( S \):

\[
P(R) - S = P(R)
\]

Now, \( |(P(R) - S) \times S| \) refers to the size of the Cartesian product of \( P(R) - S \) with \( S \). Since \( |P(R)| = 16 \) and \( |S| = 4 \):

\[
|(P(R) - S) \times S| = 16 \times 4 = 64
\]

### Final Answers:
1. \( R - T = \{6\} \)
2. \( S - P(R) = \{(2), \{2, 3, 4\}, \{2, 5, 6\}\} \)
3. \( P(R) \cap P(T) = \{\emptyset, \{2\}, \{3\}, \{7\}, \{2, 3\}, \{2, 7\}, \{3, 7\}, \{2, 3, 7\}\} \)
4. \( (R - \{3, 4, 5, 6\}) \times (Z - T) = \{(2, 4), (2, 5), (2, 6), (7, 4), (7, 5), (7, 6)\} \)
5. \( |(P(R) - S) \times S| = 64 \)

Would you like further details or explanations on any of the steps?

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Here are 5 related questions:

1. How would the solution change if \( R \) included the number 5?
2. What is the relationship between the cardinalities of \( R \), \( S \), and \( T \)?
3. How do power sets help in understanding the structure of subsets within a given set?
4. If \( S \) were a subset of \( P(R) \), what would \( S - P(R) \) be?
5. What is the significance of the Cartesian product in set theory?

**Tip:** Understanding the relationship between different set operations can help in visualizing complex problems involving multiple sets.

Explore set theory operations including set differences, intersections, power sets, and Cartesian products. Solve specific examples involving sets R, S, and T with detailed explanations and step-by-step solutions. Suitable for advanced mathematics students.

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