Remark 1. Defining some sets. • Let R be the set of all real numbers. • A subset S of R is closed under addition if for every x,y E S we have r+y S. • A subset S of R is closed under multiplication if for every r, y E S we have r y E S. • Let S = {r ER|r 5k+1 for some k E Z}. The set of integers with remainder 1 when divided by 5. %3D 1. Complete the proof that S is closed under multiplication. Proof: Suppose that r, y E S. We want to show Since r and y are in S, we know that Consider ry= 2. Prove that S is not closed under addition.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Remark 1. Defining some sets.
• Let R be the set of all real numbers.
• A subset S of R is closed under addition if for every r, y ES we have r+ yE S.
• A subset S of R is closed under multiplication if for every r, y E S we have r y E S.
• Let S {r eR|r 5k+1 for some k e Z}. The set of integers with remainder 1 when
divided by 5.
1. Complete the proof that S is closed under multiplication.
Proof: Suppose that r, y E S. We want to show
Since r and y are in S, we know that
Consider ry =
2. Prove that S is not closed under addition.
Transcribed Image Text:Remark 1. Defining some sets. • Let R be the set of all real numbers. • A subset S of R is closed under addition if for every r, y ES we have r+ yE S. • A subset S of R is closed under multiplication if for every r, y E S we have r y E S. • Let S {r eR|r 5k+1 for some k e Z}. The set of integers with remainder 1 when divided by 5. 1. Complete the proof that S is closed under multiplication. Proof: Suppose that r, y E S. We want to show Since r and y are in S, we know that Consider ry = 2. Prove that S is not closed under addition.
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