AJAY PATHAK (MATH)

Q.3 (a) Question: Define a complete graph and find the order and size of the graph K 2025 . Definition: Complete Graph A complete graph is a simple graph in which every pair of distinct vertices is connected by exactly one edge . A complete graph with n vertices is denoted by K n . Order of the Graph: The order of a graph is the number of vertices in the graph. Order of K 2025 = 2025 Size of the Graph: The size of a graph is the number of edges. A complete graph with n vertices has: n(n − 1)/2 edges For K 2025 : = 2025 × 2024 / 2 = 2025 × 1012 = 2049300 edges Final Answer: Order of K 2025 = 2025 Size of K 2025 = 2049300 Q.3 (b) Question: Show that the proposition [(p → q) ∧ (q → r)] → (p → r) is a tautology . Method: Trut...

Q.1 (a) Prove that: (A ∩ B) ∪ (A − B) = A A ∪ (B − A) = A ∪ B Solution: (i) To prove: (A ∩ B) ∪ (A − B) = A Let x ∈ (A ∩ B) ∪ (A − B) . Then either x ∈ A ∩ B or x ∈ A − B . In both cases, x ∈ A . Hence, (A ∩ B) ∪ (A − B) ⊆ A . Now let x ∈ A . Then either x ∈ B or x ∉ B . If x ∈ B , then x ∈ A ∩ B . If x ∉ B , then x ∈ A − B . Hence, x ∈ (A ∩ B) ∪ (A − B) . Therefore, A ⊆ (A ∩ B) ∪ (A − B) . Thus, (A ∩ B) ∪ (A − B) = A. (ii) To prove: A ∪ (B − A) = A ∪ B Let x ∈ A ∪ (B − A) . Then x ∈ A or x ∈ B − A . In both cases, x ∈ A ∪ B . Hence, A ∪ (B − A) ⊆ A ∪ B . Conversely, let x ∈ A ∪ B . If x ∈ A , then x ∈ A ∪ (B − A) . If x ∈ B and x ∉ A , then x ∈ B − A . Hence, A ∪ B ⊆ A ∪ (B − A) . Therefore, A ∪ (B − A) = A ∪ B. ✔ Proved ...