) Draw Venn diagram to represent followings: (3)
i) (A

B

C)

(A

B

C)
ii) (A

B

C)

(B

C)
iii) (A

B

C)
(b) Write down suitable mathematical statement that can be represented by the following symbolic
properties. (4)
i) (

x) (

y) (

z)P
ii) 
(z) (

y) (

z)Q

### (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 \(z\).

---

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.

Draw Venn diagram to represent followings: (3)
i) (A ∩ B ∩ C) ∩ (A ∩ B ∩ C)
ii) (A ∩ B ∩ C) ∪ (B ∪ C)
iii) (A ∩ B) ∪ C
(b) Write down suitable mathematical statement that can be represented by the following symbolic properties. (4)
i) (∃x) (∃y) (∃z)P
ii) ∃(z) (∃y) (∀z)Q

This page explains how to represent set operations using Venn diagrams and explores symbolic logic with existential and universal quantifiers. Detailed breakdowns of set theory concepts are provided, including how to interpret (A ∩ B ∩ C) ∪ (B ∪ C) and symbolic statements like (∃x) (∃y) (∃z)P.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation