(a) Drawing Venn Diagrams to Represent the Given Sets:

To represent these set operations with Venn diagrams, we break them down:

i) $(A \cap B \cap C) \cap (A \cap B \cap C)$

This is simply the intersection of $A \cap B \cap C$ with itself, which means it's just $A \cap B \cap C$. The region is where all three sets $A$, $B$, and $C$ overlap.

• Venn Diagram Interpretation: Highlight the intersection of all three sets $A$, $B$, and $C$.

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

This means the union of the triple intersection $A \cap B \cap C$ and the union of $B$ and $C$.

• Venn Diagram Interpretation: Highlight the area where $A \cap B \cap C$ (triple intersection) occurs and include all of the areas covered by $B \cup C$.

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

This is the union of the intersection $A \cap B$ and set $C$.

• Venn Diagram Interpretation: Highlight the region where $A$ and $B$ intersect, along with the entirety of set $C$.

(b) Suitable Mathematical Statements Represented by the Given Properties

i) $(\exists x)(\exists y)(\exists z)P$

This statement means "There exist $x$, $y$, and $z$ such that the predicate $P$ is true for those values."

• Interpretation: There are specific values of $x$, $y$, and $z$ for which $P(x, y, z)$ holds true.

ii) $\exists(z)(\exists y)(\forall z) Q$

This statement means "There exists a value $z$ such that for all values of $z$ and some $y$, the predicate $Q$ is true."

• Interpretation: There is at least one $z$ and a corresponding $y$ such that $Q(z, y)$ is true for all values of $z$.

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Would you like to see the Venn diagrams drawn or further explanations on the symbolic logic?

5 Related Questions:

1. How does the union of two sets differ from their intersection in a Venn diagram?
2. What is the significance of existential and universal quantifiers in mathematical logic?
3. How would the Venn diagram look if we removed one set, such as $C$, in part (iii)?
4. How can you simplify the expression $(A \cap B) \cap (A \cup C)$?
5. What would happen if the set $A$ were disjoint from $B$ in these examples?

Tip:

When working with existential quantifiers like $(\exists)$, remember that they assert the existence of at least one element that satisfies the condition, while universal quantifiers $(\forall)$ apply to all elements in the domain.