Question

Use mathematical induction in Exercises $38-46$ to prove results about sets. Prove that if $A_{1}, A_{2}, \ldots, A_{n}$ and $B_{1}, B_{2}, \ldots, B_{n}$ are sets such that $A_{j} \subseteq B_{j}$ for $j=1,2, \ldots, n,$ then $$ \bigcup_{j=1}^{n} A_{j} \subseteq \bigcup_{j=1}^{n} B_{j} $$

Name: Problem 38, Chapter 5: Discrete Mathematics and its Applications
Uploaded: 2019-08-30T14:52:47-08:00
Duration: 2 min 54 s
Channel: Prathan Jarupoonphol
Description: Use mathematical induction in Exercises 38-46 to prove results about sets. Prove that if A1, A2, …, An and B1, B2, …, Bn are sets such that Aj ⊆Bj for j=1,2, …, n, then ⋃j=1^n Aj ⊆⋃j=1^n Bj

   Use mathematical induction in Exercises $38-46$ to prove results about sets.
Prove that if $A_{1}, A_{2}, \ldots, A_{n}$ and $B_{1}, B_{2}, \ldots, B_{n}$ are sets such that $A_{j} \subseteq B_{j}$ for $j=1,2, \ldots, n,$ then
$$
\bigcup_{j=1}^{n} A_{j} \subseteq \bigcup_{j=1}^{n} B_{j}
$$

Discrete Mathematics and its Applications

Kenneth Rosen 8th Edition

Chapter 5, Problem 38

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Solved by Expert Prathan Jarupoonphol

08/30/2019

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We need to show that $A_1 \subseteq B_1$. This is given in the problem statement, so the base case holds. Show more…

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Use mathematical induction in Exercises $38-46$ to prove results about sets. Prove that if $A_{1}, A_{2}, \ldots, A_{n}$ and $B_{1}, B_{2}, \ldots, B_{n}$ are sets such that $A_{j} \subseteq B_{j}$ for $j=1,2, \ldots, n,$ then $$ \bigcup_{j=1}^{n} A_{j} \subseteq \bigcup_{j=1}^{n} B_{j} $$

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Key Concepts

Union of Sets

The union of a collection of sets is the set of all elements that belong to at least one of the sets. This concept is fundamental when working with collections of sets and understanding how properties like subset relations extend to unions of sets.

Mathematical Induction

Mathematical induction is a proof method used to establish that a given statement holds for all natural numbers. It involves proving the base case and then showing that if the statement is true for an arbitrary natural number n, it must also be true for n + 1.

Subset Relations

The subset relation, denoted by '?', indicates that all elements of one set are also elements of another. Understanding this concept is crucial, especially when comparing unions or intersections of sets.

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