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This assignment for COMSATS University Islamabad’s course, "Discrete Structures," has several questions involving sets, relations, and operations. Here is a general approach to solving each question:

1. **Set Operations with Venn Diagrams**:
   - Use set notation definitions to answer questions based on set intersections, unions, and complements.
   - Represent each query with a Venn diagram for clarity.

2. **Finding Set Intersections**:
   - For question #2, identify pairs \( a \) and \( b \) values that satisfy \( A \cap B = \emptyset \), considering conditions on the universal set \( U \), and sets \( A \) and \( B \).

3. **Cricket Club Problem (Set Cardinality)**:
   - Apply the principle of inclusion-exclusion to determine member counts in subsets like "only batsmen" or "neither batsmen nor bowlers."
   - Set up equations based on provided values for \( n(U) \), \( n(X) \), \( n(Y) \), etc., to solve for unknowns \( b \) and \( d \).

4. **Interval Notations on a Number Line**:
   - Draw number lines for given intervals and illustrate each set operation (union, intersection, complement).
   - Write expressions for each result.

5. **Medical Test Venn Diagram**:
   - Create a Venn diagram using the data for elderly men with specific diseases. Determine the counts in each subset by solving the provided conditions.

6. **Set Identity Proof**:
   - Prove the identity \( (A - B) \cup (B - C) = (A \cup B) - (B \cap C) \) using different methods: membership table, Venn diagram, laws of set theory, and an element-based argument.

7. **Relations Representations (Arrow Diagram, Digraph, Matrix)**:
   - Create arrow diagrams, directed graphs, and matrices for given relations.
   - Analyze reflexive, symmetric, and transitive properties.

8. **Four-Bit String Relation**:
   - Determine if the relation \( R \) (based on two-bit substrings) is reflexive, symmetric, transitive, antisymmetric, equivalence, or partial order.

9. **Subset Relations and Equivalence Relation**:
   - Verify if the relation \( ARB \) on power sets of \( X \) is an equivalence relation by testing properties.

10. **Partial Order for Tasks**:
    - Show that the tasks form a partial order by verifying reflexivity, antisymmetry, and transitivity.

Would you like a detailed solution for a specific question, or do you need guidance on the entire assignment? Here are some related questions to deepen understanding:

1. How does the inclusion-exclusion principle help in calculating set cardinalities?
2. Why do certain set operations, like intersections and unions, follow specific laws in set theory?
3. What properties make a relation an equivalence or a partial order?
4. How can Venn diagrams simplify solving complex set operations?
5. What role does a matrix play in representing relations on a finite set?

**Tip:** Practice with Venn diagrams and set identities helps build a solid foundation in discrete math concepts.

Find solutions to the discrete structures assignment for COMSATS University. Covers concepts like set theory, relations, and operations, including Venn diagrams, set identities, interval notations, and properties of relations such as reflexivity and transitivity. Suitable for first-year undergraduate students.

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