Question 1 Lets sets $$ A=\{x \mid x \in 2 Z\}, B=\{2,5,7,8,10\}, C=\{2,8,10\} $$ and $$ D=\{2,\{5,7,8\}, 10 $$ Complete the following by using the notation $$ \in $$ or $$ \subseteq $$ : 1. 100 $$ \qquad $$ A 2. $$ C $$ $$ \qquad $$ A 3. $$ C $$ $$ \qquad $$ B 4. $$ C $$ $$ \qquad $$ D 5. $$ \{5,7,8\} $$ $$ \qquad $$ B Question 2 Using the sets from Question 1 above, find the following: 1. $$ B \cup C $$ 2. $$ A \cap C $$ 3. $$ C \cap D $$ 4. $$ B \cap C^{\prime} $$

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1. For each relation, we must decide whether the left side is an element of the right side or a subset of the right side.

- $$ A=\{ x \mid x\in2\mathbb{Z}\} $$ is the set of all even integers.
   - $$ B=\{2,5,7,8,10\} $$
   - $$ C=\{2,8,10\} $$
   - $$ D=\{2,\{5,7,8\},10\} $$  
     Note that the second element of $$ D $$ is the set $$ \{5,7,8\} $$, not the individual numbers.

2. We now complete each relation:

1. $$ 100 \ \boxed{\in} \ A $$  
      Since $$ 100 $$ is an even number, $$ 100\in A $$.

2. $$ C \ \boxed{\subseteq} \ A $$  
      Every element of $$ C $$ (namely, $$ 2 $$, $$ 8 $$, and $$ 10 $$) is even, so $$ C\subseteq A $$.

3. $$ C \ \boxed{\subseteq} \ B $$  
      The elements of $$ C $$ are $$ 2 $$, $$ 8 $$, and $$ 10 $$, all of which are in $$ B $$. Thus, $$ C\subseteq B $$.

4. $$ C \ \boxed{\not\subseteq} \ D $$  
      Although $$ D $$ contains $$ 2 $$ and $$ 10 $$, it does not contain $$ 8 $$ (its second element is the set $$ \{5,7,8\} $$ which is not the same as $$ 8 $$). Thus, $$ C $$ is not a subset of $$ D $$. Also, $$ C $$ is not an element of $$ D $$.

5. $$ \{5,7,8\} \ \boxed{\subseteq} \ B $$  
      The set $$ \{5,7,8\} $$ has all its elements in $$ B $$, so $$ \{5,7,8\}\subseteq B $$.

3. Now, we solve Question 2 using the given sets.

1. $$ B\cup C $$  
      We have
      $$
      B=\{2,5,7,8,10\} \quad\text{and}\quad C=\{2,8,10\}.
      $$
      Their union is
      $$
      B\cup C=\{2,5,7,8,10\}.
      $$

2. $$ A\cap C $$  
      Since $$ A $$ is the set of even numbers and $$ C=\{2,8,10\} $$,
      $$
      A\cap C=\{2,8,10\}.
      $$

3. $$ C\cap D $$  
      The set
      $$
      D=\{2,\{5,7,8\},10\}
      $$
      has elements $$ 2 $$ and $$ 10 $$ that are also in $$ C $$. The element $$ 8 $$ is not in $$ D $$ (because $$ 8 \neq \{5,7,8\} $$). Thus,
      $$
      C\cap D=\{2,10\}.
      $$

4. $$ B\cap C^{\prime} $$  
      Here, $$ C^{\prime} $$ denotes the complement of $$ C $$ relative to some universal set. In this context (since we are intersecting with $$ B $$), we take the relative complement in $$ B $$; that is, we remove the elements of $$ C $$ from $$ B $$.

Since
      $$
      B=\{2,5,7,8,10\} \quad\text{and}\quad C=\{2,8,10\},
      $$
      we have
      $$
      B\cap C^{\prime}=\{5,7\}.
      $$

Final Answers:

1. $$ 100 \in A $$  
   $$ C \subseteq A $$  
   $$ C \subseteq B $$  
   $$ C \not\subseteq D $$  
   $$ \{5,7,8\}\subseteq B $$

2. $$ B\cup C=\{2,5,7,8,10\} $$  
   $$ A\cap C=\{2,8,10\} $$  
   $$ C\cap D=\{2,10\} $$  
   $$ B\cap C^{\prime}=\{5,7\} $$
1. $$ 100 \in A $$ $$ C \subseteq A $$ $$ C \subseteq B $$ $$ C \not\subseteq D $$ $$ \{5,7,8\} \subseteq B $$ 2. $$ B \cup C = \{2,5,7,8,10\} $$ $$ A \cap C = \{2,8,10\} $$ $$ C \cap D = \{2,10\} $$ $$ B \cap C' = \{5,7\} $$

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