Question 3 a) Suppose $$ E=\{ $$ cat, dog, rabbit, mouse $$ \}, F=\{ $$ dog, cow, duck, pig, rabbit $$ \} $$ mouse, python $$ \} $$ Find the following: 1. $$ (E \cap F) \cup G $$ 2. $$ E \cap(F \cup G) $$ 3. $$ (E \cap F)^{\prime} \cap G $$ b) If $$ \mathbf{U}=\{2,3,4,5,6,7,8,9,10,11\}, A=\{3,5,7,9,11\} $$ and $$ B=\{7,8,9,10,11\} $$ 1. Then find $$ (\mathbf{U} \cap A) \cup(\mathbf{U} \cap B) $$, 2. and also find $$ (A-B)^{\prime} $$. Question 4 By the use of a venn diagram, answer the following questions: 1. $$ A \cup B $$ 2. $$ A \cap B $$ 3. $$ A \cap B $$ 4. $$ B \cap A $$ 5. $$ (A \cup B)^{\prime} $$

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**3(a)**  
Let  
$$
E=\{\text{cat},\text{dog},\text{rabbit},\text{mouse}\},\quad F=\{\text{dog},\text{cow},\text{duck},\text{pig},\text{rabbit}\},\quad G=\{\text{mouse},\text{python}\}.
$$

1. First, find  
   $$
   E\cap F=\{\text{dog},\text{rabbit}\}
   $$  
   Then,  
   $$
   (E\cap F)\cup G=\{\text{dog},\text{rabbit}\}\cup\{\text{mouse},\text{python}\}=\{\text{dog},\text{rabbit},\text{mouse},\text{python}\}.
   $$
   
2. First, form the union  
   $$
   F\cup G=\{\text{dog},\text{cow},\text{duck},\text{pig},\text{rabbit}\}\cup\{\text{mouse},\text{python}\}=\{\text{dog},\text{cow},\text{duck},\text{pig},\text{rabbit},\text{mouse},\text{python}\}.
   $$  
   Then, take the intersection with $$ E $$:  
   $$
   E\cap (F\cup G)=\{\text{cat},\text{dog},\text{rabbit},\text{mouse}\}\cap\{\text{dog},\text{cow},\text{duck},\text{pig},\text{rabbit},\text{mouse},\text{python}\}
   $$
   which gives  
   $$
   E\cap (F\cup G)=\{\text{dog},\text{rabbit},\text{mouse}\}.
   $$
   
3. First, recall that  
   $$
   E\cap F=\{\text{dog},\text{rabbit}\}.
   $$  
   To form the complement, we assume the universal set is  
   $$
   U=E\cup F\cup G=\{\text{cat},\text{dog},\text{rabbit},\text{mouse},\text{cow},\text{duck},\text{pig},\text{python}\}.
   $$  
   Then,  
   $$
   (E\cap F)'=U\setminus\{\text{dog},\text{rabbit}\}=\{\text{cat},\text{mouse},\text{cow},\text{duck},\text{pig},\text{python}\}.
   $$  
   Now,  
   $$
   (E\cap F)'\cap G=\{\text{cat},\text{mouse},\text{cow},\text{duck},\text{pig},\text{python}\}\cap\{\text{mouse},\text{python}\}=\{\text{mouse},\text{python}\}.
   $$

---

**3(b)**  
Given  
$$
\mathbf{U}=\{2,3,4,5,6,7,8,9,10,11\},\quad A=\{3,5,7,9,11\},\quad B=\{7,8,9,10,11\}.
$$

1. Notice that $$ A\subset \mathbf{U} $$ and $$ B\subset \mathbf{U} $$, so  
   $$
   \mathbf{U}\cap A=A,\quad \mathbf{U}\cap B=B.
   $$  
   Then,  
   $$
   (\mathbf{U}\cap A)\cup (\mathbf{U}\cap B)=A\cup B=\{3,5,7,9,11\}\cup\{7,8,9,10,11\}=\{3,5,7,8,9,10,11\}.
   $$

2. First, find the set difference  
   $$
   A-B=\{x\in A: x\notin B\}.
   $$  
   Since  
   $$
   A=\{3,5,7,9,11\}\quad \text{and} \quad B=\{7,8,9,10,11\},
   $$  
   we have  
   $$
   A-B=\{3,5\}.
   $$  
   The complement $$ (A-B)' $$ (assuming the universal set is $$\mathbf{U}$$) is  
   $$
   (A-B)'=\mathbf{U}\setminus\{3,5\}=\{2,4,6,7,8,9,10,11\}.
   $$

---

**4**  
Let $$ A $$ and $$ B $$ be two sets represented by overlapping circles in a Venn diagram. Then:

1. $$ A\cup B $$ represents all elements in $$ A $$ or $$ B $$ (or both).  
2. $$ A\cap B $$ represents the elements common to both $$ A $$ and $$ B $$.  
3. $$ A\cap B $$ (repeated) is the same as in item 2.  
4. $$ B\cap A $$ is identical to $$ A\cap B $$ (since intersection is commutative).  
5. $$ (A\cup B)' $$ represents all elements not in $$ A $$ and not in $$ B $$ (i.e., the region outside both circles).

Each of these expressions corresponds to the respective regions in the Venn diagram.
**3(a)** 1. $$ (E \cap F) \cup G = \{\text{dog}, \text{rabbit}, \text{mouse}, \text{python}\} $$ 2. $$ E \cap (F \cup G) = \{\text{dog}, \text{rabbit}, \text{mouse}\} $$ 3. $$ (E \cap F)' \cap G = \{\text{mouse}, \text{python}\} $$ **3(b)** 1. $$ (\mathbf{U} \cap A) \cup (\mathbf{U} \cap B) = \{3,5,7,8,9,10,11\} $$ 2. $$ (A - B)' = \{2,4,6,7,8,9,10,11\} $$ **4** 1. $$ A \cup B $$: All elements in $$ A $$ or $$ B $$ (or both). 2. $$ A \cap B $$: Elements common to both $$ A $$ and $$ B $$. 3. $$ A \cap B $$: Same as item 2. 4. $$ B \cap A $$: Same as $$ A \cap B $$. 5. $$ (A \cup B)' $$: Elements not in $$ A $$ or $$ B $$.

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