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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**Question 3**

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

1. To find $$\,(E \cap F) \cup G\,$$:

- First, compute $$E \cap F$$:  
     The common elements of $$E$$ and $$F$$ are  
     $$
     E \cap F=\{\text{dog, rabbit}\}.
     $$
     
   - Now, take the union with $$G$$:  
     $$
     (E \cap F) \cup G = \{\text{dog, rabbit}\} \cup \{\text{mouse, python}\} = \{\text{dog, rabbit, mouse, python}\}.
     $$

2. To find $$\,E \cap (F \cup G)\,$$:

- First, compute $$F \cup G$$:  
     $$
     F \cup G = \{\text{dog, cow, duck, pig, rabbit}\} \cup \{\text{mouse, python}\} = \{\text{dog, cow, duck, pig, rabbit, mouse, python}\}.
     $$
     
   - Now, intersect with $$E$$:  
     The elements of $$E$$ are $$\{\text{cat, dog, rabbit, mouse}\}$$ so we have  
     $$
     E \cap (F \cup G)=\{\text{dog, rabbit, mouse}\}.
     $$

3. To find $$\,(E \cap F)^{\prime} \cap G\,$$:

Assume that the complement $$(E \cap F)'$$ is taken relative to the universal set  
   $$
   U = E \cup F \cup G = \{\text{cat, dog, rabbit, mouse, cow, duck, pig, python}\}.
   $$
   
   - We already found  
     $$
     E \cap F=\{\text{dog, rabbit}\}.
     $$
     
   - Then the complement is  
     $$
     (E \cap F)' = U \setminus \{\text{dog, rabbit}\} = \{\text{cat, mouse, cow, duck, pig, python}\}.
     $$
     
   - Now, intersect with $$G$$:  
     $$
     (E \cap F)' \cap G = \{\text{cat, mouse, cow, duck, pig, python}\} \cap \{\text{mouse, python}\} = \{\text{mouse, python}\}.
     $$

**(b)**  
We are 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. To find $$\,(\mathbf{U} \cap A) \cup (\mathbf{U} \cap B)\,$$:

Since $$A$$ and $$B$$ are subsets of $$\mathbf{U}$$, we have  
   $$
   \mathbf{U} \cap A = A, \quad \mathbf{U} \cap B = B.
   $$
   Their union is  
   $$
   A \cup B = \{3,5,7,9,11\} \cup \{7,8,9,10,11\} = \{3,5,7,8,9,10,11\}.
   $$

2. To find $$\,(A-B)^{\prime}\,$$:

- First, find $$A - B$$ (elements in $$A$$ not in $$B$$):  
     $$
     A - B = \{3,5,7,9,11\} \setminus \{7,8,9,10,11\} = \{3,5\}.
     $$
     
   - The complement is taken relative to $$\mathbf{U}$$. Thus,  
     $$
     (A - B)^{\prime} = \mathbf{U} \setminus \{3,5\} = \{2,4,6,7,8,9,10,11\}.
     $$

---

**Question 4**

In a Venn diagram the universal set is typically represented by a rectangle and the sets $$A$$ and $$B$$ by overlapping circles. The answers are as follows:

1. $$\,A \cup B\,$$: This represents all elements that are in $$A$$ or in $$B$$ (or in both). In the Venn diagram, this is the entire area of both circles.

2. $$\,A \cap B\,$$: This represents the elements common to both $$A$$ and $$B$$. In the Venn diagram, this is the overlapping region of the two circles.

3. $$\,A \cap B\,$$: (Same as above.) It is the overlapping region of $$A$$ and $$B$$.

4. $$\,B \cap A\,$$: (Again, same as $$A \cap B$$.) The intersection is commutative, so its representation is the same overlapping region.

5. $$\,(A \cup B)^{\prime}\,$$: This represents the elements that are neither in $$A$$ nor in $$B$$. In the Venn diagram, this is the area of the rectangle outside both circles.
**Question 3** **(a)** Given: $$ E = \{\text{cat, dog, rabbit, mouse}\},\quad F = \{\text{dog, cow, duck, pig, rabbit}\},\quad G = \{\text{mouse, python}\} $$ 1. $$(E \cap F) \cup G = \{\text{dog, rabbit, mouse, python}\}$$ 2. $$E \cap (F \cup G) = \{\text{dog, rabbit, mouse}\}$$ 3. $$(E \cap F)' \cap G = \{\text{mouse, python}\}$$ **(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. $$(\mathbf{U} \cap A) \cup (\mathbf{U} \cap B) = \{3,5,7,8,9,10,11\}$$ 2. $$(A - B)' = \{2,4,6,7,8,9,10,11\}$$ --- **Question 4** In a Venn diagram: 1. $$A \cup B$$ includes all elements in $$A$$ or $$B$$ (or both). 2. $$A \cap B$$ includes elements common to both $$A$$ and $$B$$. 3. $$A \cap B$$ (same as above). 4. $$B \cap A$$ (same as $$A \cap B$$). 5. $$(A \cup B)'$$ includes elements not in $$A$$ or $$B$$.

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