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Counting Integers Divisible by 5 but Not by 7
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Okay class, today we're looking at the set S, which includes integers like 0, 5, -5, 10, and so forth. However, we exclude integers that are divisible by both 5 and 7. Can anyone give me an example of an integer we should exclude?
Yes! 35 is an example since it is divisible by both 5 and 7.
Exactly! Now, why do you think set S is considered infinite?
Because we can keep adding more multiples of 5 in both positive and negative directions indefinitely.
Right! Now let’s consider how we can list or enumerate these elements. What approach can we take?
We should start with 0, then include +5, and -5, followed by +10 and -10.
Great! However, remember, we cannot simply list positive numbers followed by negatives because we could get stuck. We need a balanced approach. Review this approach: 0, +5, -5, +10, -10... How does it look now?
It looks organized, and all elements are accounted for!
Perfect! So, we conclude that set S is infinite and countable. This means we can enumerate the elements methodically without missing any.
Countability of Real Numbers with Decimal Representation of Only 1s
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Let's delve into another interesting set. This one consists of all real numbers whose decimal representation includes only the digit '1'. Can anyone think of examples from this set?
How about 0.1, 1.11, and the number 1?
Great examples! Now, what do you think about the countability of this set?
I think it’s countable since we can arrange them in lists based on their patterns.
Exactly! We can break this set down into smaller parts; let’s denote these as S1, S2, and so on. Each subset can be enumerated easily. Could someone explain how that works?
S1 could include numbers like 0.1, 0.11, and recurring 1s only after the decimal! And S2 could feature numbers starting with 1 followed by 1s.
Right again! When we take the union of all our countable subsets, the overall set S remains countable. Each S_n contributes to the larger set, allowing for orderly enumeration.
So essentially, countable unions of countable sets lead us to the conclusion that S is countably infinite?
Precisely! So we conclude that even though real numbers seem inherently uncountable, we can identify countable subsets within them.
Introduction & Overview
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Quick Overview
Standard
We explore the countability of two specific sets: integers divisible by 5 but not by 7, and real numbers whose decimal representation consists only of 1s. The discussion involves the process of enumeration and recognizing infinite patterns within these sets.
Detailed
Detailed Summary
In this section, we focus on the analysis of countability in two specific contexts. The first part evaluates the set of integers divisible by 5 but not by 7. This set, denoted as S, includes integers such as 0, ±5, ±10, and so forth, while excluding any integer that is a multiple of both 5 and 7. The conclusion is that while set S is infinite, it is still countable because we can enumerate its elements in a valid sequence, accounting for positive and negative values.
The second part investigates the set of real numbers whose decimal representations consist solely of the digit '1'. This set proves to be countable as it can be constructed from several countable subsets. The total union of those subsets allows us to list all qualifying real numbers, indicating that set S is indeed countable. This guided exploration reinforces the understanding of different types of infinities within the context of mathematical sets.
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Part (a): Set of Integers Divisible by 5 but not by 7
Chapter 1 of 2
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Chapter Content
In part a, you want to find out whether the set of integers divisible by 5 but not by 7 is countable or not. So first of all what exactly is this set so let me call, denote this set by S. So the set S will have the number 0 it will have +5 – 5 it will have +10 – 10 and so on. It would not have 35 it would not have + 35 it would not have -35 because 35 is divisible by 5. But it is also divisible by 7 so are not supposed to include multiples of 7. We are not supposed to include multiples of both 5 and 7; so now we want to show whether the set S is countable or not. So definitely S is an infinite set but the question is can we enumerate the elements of the set S. And it is easy see that we can always list down the elements of the set according to their absolute values. And if I focus on the absolute values then both +x and –x will take the same absolute value x. So what I will do is in the listing I will write down +x and –x so basically what I am saying is that my listing here is 0 followed by +5 followed by -5 then +10 followed by -10 and so on. You cannot do the following: you cannot say that first list down all the positive things up to infinity and then followed by the minus things. This is not a valid listing because if we do this then we do not know whether we will come back ever and start listing down the negative numbers here. Because when we start going towards the positive multiples of 5 in infinity we will never stop and there is no coming back. So that is why it is only the first listing which is a valid listing not the second one.
Detailed Explanation
The set S contains integers that are divisible by 5 but not by 7. This means that the set includes values like 0, 5, 10, -5, -10, etc., but excludes multiples of 35 (5 and 7's product). Since this set includes both positive and negative values and continues indefinitely in both directions, it is infinite. We can enumerate the members of S by listing them as follows: 0, 5, -5, 10, -10, etc. Importantly, while we can create a sequence, we cannot just list all the positive integers first and then the negative ones because that would make us lose the order needed to cover all elements. Thus, S is countable, meaning we can list its integers in a structured manner.
Examples & Analogies
Think of it like organizing a collection of books. You have books that belong to specific genres. Imagine if you decide to list them by genre, you would list all the books of one genre before moving onto another. Now, if you instead said, 'I will list all the fiction books first and then the non-fiction books next,' you might end up forgetting some books or leaving them out because you get caught up in one genre. The same principle applies here when it comes to ensuring we enumerate S correctly.
Part (b): Real Numbers with Decimal Representation of Only 1s
Chapter 2 of 2
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Chapter Content
Part B we want to find out whether the set of all real numbers whose decimal representation consist of only 1s is countable or not. So remember the set of real numbers is uncountable we proved that but here we are not focusing on the set of all real numbers. We are focusing only on the real numbers whose decimal representation have only 1s and no other digit. And it turns out that this set is countable because I can view it as union of several countable sets okay. So let S denotes the real numbers consisting of only 1s where you do not have anything before the decimal point. And after the decimal point you list down all the numbers starting with real number where you have recurring 1s that means you have a series of infinite ones and occurring again and again. Followed, by the real number 0.1 followed by real number 0.11 followed by real number 0.111 and so on. Remember 0.1 is not same as the real number where 1 is recurring. 0.1 you do not have anything after the first 1 here. But 0.1 with the recurring one denotes this real number which is clearly different from 0.1. So these are the elements in my set S and I have a valid listing here.
Detailed Explanation
This part focuses on determining whether the subset of real numbers that only have the digit 1 in their decimal representation is countable. The creator of the set, S, breaks it down into smaller sets, where each smaller set corresponds to numbers with a different number of 1s before the decimal. Each of these smaller sets is countable, and by taking their union, we still have a countable set since the union of countable sets remains countable. This shows that even though our focus is limited to numbers with 1s, they can still be listed systematically.
Examples & Analogies
Imagine a bakery that only bakes pastries with a single type of topping, say chocolate. If the bakery decides to make different styles of pastries with that one topping, it can create a systematic layout. Think of these styles as different countable sets. If there’s a list for each style, then the overall collection of chocolate pastries is organized and countable. It's just like listing numbers based on the number of 1s in their decimal representation.
Key Concepts
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Countable Sets: These are sets that can be enumerated or listed without missing any elements.
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Infinite Sets: Sets containing an endless number of elements, such as integers or real numbers.
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Enumerating a Set: The process of listing elements systematically to illustrate their countability.
Examples & Applications
The set of all integers divisible by 5 but not by 7 includes numbers like 0, ±5, ±10, etc.
The set of real numbers with decimal representations only showing 1s includes 0.1, 1.1, 1.11, etc.
Memory Aids
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Rhymes
Count the 5s, skip the 7s, integers go to infinity, like the way we count our blessings.
Stories
Imagine journeying on a number line, collecting all multiples of 5 while dodging the multiples of 7 like puddles in the rain.
Memory Tools
For set representations, remember: Just 1! That's all it takes to count real numbers that look the same.
Acronyms
SIX → S = Set, I = Integers, X = eXclude (7's).
Flash Cards
Glossary
- Countable Set
A set that can be placed in one-to-one correspondence with the set of positive integers, meaning its elements can be listed or enumerated.
- Infinite Set
A set that contains an endless number of elements, implying it is not finite.
- Enumerate
To list elements systematically, often in a defined order.
- Subset
A set formed from a given set, retaining some or all of its elements.
- Real Numbers
The set of numbers that include all rational and irrational numbers.
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