Question

Consider the sample space S = {copper, sodium, nitrogen, potassium, uranium, oxygen, zinc}, and the events A={copper, sodium, zinc}, B={uranium, nitrogen, potassium}, C={oxygen}.
List the elements of the sets corresponding the following events:
i) A ∪ C
ii) (A ∩ B') ∪ C'
iii) (A' ∪ B') ∩ (A' ∩ C)

Answer 1

Answer on Question #42118 – Math – Set Theory

Consider the sample space $S = \{\text{copper}, \text{sodium}, \text{nitrogen}, \text{potassium}, \text{uranium}, \text{oxygen}, \text{zinc}\}$, and the events $A = \{\text{copper}, \text{sodium}, \text{zinc}\}$, $B = \{\text{uranium}, \text{nitrogen}, \text{potassium}\}$, $C = \{\text{oxygen}\}$.

List the elements of the sets corresponding the following events:

i) $A \cup C$

ii) $(A \cap B') \cup C'$

iii) $(A' \cup B') \cap (A' \cap C)$

Solution:

Rename the elements of $S$ the figures for convenience:

- Copper => 0,

- Sodium => 1,

- Nitrogen => 2,

- Potassium => 3,

- Uranium => 4,

- Oxygen => 5,

- Zinc => 6,

Then:

$$S = \{0, 1, 2, 3, 4, 5, 6\}; A = \{0, 1, 6\}; B = \{2, 3, 4\}; C = \{5\}$$

1) $A \cup C = \{0,1,5,6\}$ or $A \cup C = \{\text{copper}, \text{sodium}, \text{oxygen}, \text{zinc}\}$

2) $B' = \{0,1,5,6\}$

$$C' = \{0, 1, 2, 3, 4, 6\}$$ $$A \cap B' = \{0, 1, 6\} = D$$ $$D \cup C' = (A \cap B') \cup C' = \{0, 1, 2, 3, 4, 6\} \text{ or } (A \cap B') \cup C' = \{\text{copper}, \text{sodium}, \text{nitrogen}, \text{potassium}, \text{uranium}, \text{zinc}\}$$

3) $A' = \{2,3,4,5\}$

$$B' = \{0, 1, 5, 6\}$$ $$A' \cup B' = \{0, 1, 2, 3, 4, 5, 6\} = S$$ $$A' \cap C = \{5\}$$ $$(A' \cup B') \cap (A' \cap C) = S \cap \{5\} = \{5\} \text{ or } (A' \cup B') \cap (A' \cap C) = \{\text{oxygen}\}$$

Answer:

1) $A \cup C = \{\text{copper}, \text{sodium}, \text{oxygen}, \text{zinc}\}$

2) $(A \cap B') \cup C' = \{\text{copper}, \text{sodium}, \text{nitrogen}, \text{potassium}, \text{uranium}, \text{zinc}\}$

3) $(A' \cup B') \cap (A' \cap C) = \{\text{oxygen}\}$

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