Exercise 3.4.4. Let A and B be non-empty families of sets. Suppose that A ⊆ B. (1) Prove that U X∈A X ⊆ U Y∈B Y. (2) Prove that ∩ X∈A X ⊆ ∩ Y∈B Y. Exercise 3.4.5. Let I be a non-empty set, and let {Ai}i∈I and {Bi}i∈I be families of sets indexed by I. Suppose that Ai ⊆ Bi for all i ∈ I. (1) Prove that U i∈I Ai ⊆ U i∈I Bi. (2) Prove that ∩ i∈I Ai ⊆ ∩ i∈I Bi.

Exercise 3.4.4. Let A and B be non-empty families of sets. Suppose that A ⊆ B. (1) Prove that U X∈A X ⊆ U Y∈B Y. (2) Prove that ∩ X∈A X ⊆ ∩ Y∈B Y. Exercise 3.4.5. Let I be a non-empty set, and let {Ai}i∈I and {Bi}i∈I be families of sets indexed by I. Suppose that Ai ⊆ Bi for all i ∈ I. (1) Prove that U i∈I Ai ⊆ U i∈I Bi. (2) Prove that ∩ i∈I Ai ⊆ ∩ i∈I Bi.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Exercise 3.4.4. Let A and B be non-empty families of sets. Suppose that A ⊆ B.

(1) Prove that U X∈A X ⊆ U Y∈B Y. (2) Prove that ∩ X∈A X ⊆ ∩ Y∈B Y.

Exercise 3.4.5. Let I be a non-empty set, and let {Ai}i∈I and {Bi}i∈I be families of sets indexed by I. Suppose that Ai ⊆ Bi for all i ∈ I.

(1) Prove that U i∈I Ai ⊆ U i∈I Bi. (2) Prove that ∩ i∈I Ai ⊆ ∩ i∈I Bi.

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