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.