If X and Y are sets, prove thouroughly that X⊆Y if and only if X∩Y = X.
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If X and Y are sets, prove thouroughly that X⊆Y if and only if X∩Y = X.
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- . Let Q, R and S be sets. Show that (R – Q) ∪ (S – Q) = (R ∪ S) – Q.Show that if A and B are sets with A ⊆ B then A ∪ B = B. (Hint: Show that x ∈ A ∪ B implies x ∈ B and vice versa.)2. Consider the given sets A and B. Show a possible bijection for the function f, where f: A-B, and define f for that bijection. a. A= {1, 2, 3, 4, ..., 10} and B = {2, 5, 10, 17, ...,101} •••. 3 b. A= {0, 1}° and B = {1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111}
- [Introduction to the Design and Analysis of Algorithms, 3rd Edition] Maxima search. A point (xi, yi) in the Cartesian plane is said to be dominated by point (xj , yj ) if xi ≤ xj and yi ≤ yj with at least one of the two inequalities being strict. Given a set of n points, one of them is said to be a maximum of the set if it is not dominated by any other point in the set. For example, in the figure below, all the maximum points of the set are circled. Design an efficient algorithm for finding all the maximum points of a given set of n points in the Cartesian plane. What is the time efficiency class of your algorithm?Let A, B,C be sets. Prove or disprove А — (А — В) %3D AnВUse a direct proof technique to prove the following theorems: For all integers x and y, x2 + y2 − 3 is not divisible by 4. The sets A, B, and C are arbitrary subsets of some universal set U. Prove that (C − A) ∪ (C − B) = C − (A ∩ B).
- Prove that equivalences of sets are transitive, for example if X ~ Y and Y ~ Z then X ~ Z.Given A and B are sets, prove that (A ⋃ B) ⋃ C = A ⋃ (B ⋃ C) by showing (A ⋃ B) ⋃ C ⊆ A ⋃ (B ⋃ C) and A ⋃ (B ⋃ C) ⊆ (A ⋃ B) ⋃ C. (You cannot use the Associative Laws for Sets since this is what you are proving.)Suppose we have the following sets: P, Q, and R. Prove or disprove that if for all x, (x ∈ P) → ((x ∈ Q) → (x ∈ R)), then P ⋂ Q ⊆ R