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Prove that for all natural numbers n and m, if In Im then n = m Hint: Use induction 0n n.)-dditional InformationDefinition 8.1.1. Suppose and are sets. We'1l s...

Question

Prove that for all natural numbers n and m, if In Im then n = m Hint: Use induction 0n n.)-dditional InformationDefinition 8.1.1. Suppose and are sets. We'1l say that equinumerous with B if there is a function f: A B that is one-tO-one and onto. We 'Il write A B to indicate that A is equinumerous with B. For each natural number n, let In = {i €zt | i < n}. A set A is called finite if there is a natural number n such that I, A. Otherwise, A is infinite.

Prove that for all natural numbers n and m, if In Im then n = m Hint: Use induction 0n n.) -dditional Information Definition 8.1.1. Suppose and are sets. We'1l say that equinumerous with B if there is a function f: A B that is one-tO-one and onto. We 'Il write A B to indicate that A is equinumerous with B. For each natural number n, let In = {i €zt | i < n}. A set A is called finite if there is a natural number n such that I, A. Otherwise, A is infinite.



Answers

Show that if $A$ is an infinite set, then it contains a countably infinite subset.

Let's call ass the sad containing 0123456789 Now let's say, uh, real number between zero and one and write it as zero point the one b two b three and so forth where the eyes are its digit which, of course, our numbers between 09 and we define the function f of X that goes from the positive integers tow us. So the set of digits that sends a number and thio the digit of X. So, for instance, if our number axes 0.10279 Then we did find the function of of ex in a way that the maps one toe the first digit of acts which is one for backs maps to to the second digital acts, which is zero The maps three to the third digital backs, which is a two four to the fourth digit, which is a seven five to that fifth digit which is a nine and so forth so we can define a function g. There goes from the interval 01 to the set off functions from that plus, So the set of Regent s, which sends a number X to the function that we did find up there ever backs, which, of course, depends on X. We want to show that these are my objection. So first of all, I want to show the G is injected. So let's assume that G of X is equal to y By our definition of G, that means that the functions f of X and if so, why are the same. But then equality between functions means that the functions go inside wherever they're computed. So f of X is equal to f y ove n for every possible end. But this means that the digit off the number X is equal to the end digit off the number. Why? Because these are we to find the functions ever vex, and f y an end is any number from one all the way up to infinity. So that means that every digit of acts you need to get him expansion is equal to every digit really corresponding digital. Why in the decimal expansion and therefore X and y out of the same. And now, to show the G's objective, find the function I mean, take a function of from the positive integers West. So any function of and the final reel number X bye decimal expansion zero point the one detour the three and so forth whereby construction it's digit is and reached. The end is given by our function off completely then, but then his number X must be in 01 and Jax is indeed the function after we started with Now. The observation here is that we need to include zero and one in the evening Indian Trimmel because there's an issue with the decimal expansion, which is if we take number 0.9999 and only nine. That's the same as one, because the decimal expansion well two different decimal expansions can be can represent the same number so far into the number one. So 1.0 can be written also at 0.9 periodic. So we need to include one and also any to Groot zero. Because we could have the function half the goes well there as takes all all the values zeros, and so we would have 0.0 So that's why we need it included 01 in the interval but it means a G is by Jack is by objective and therefore the set of functions from that plus to us is uncomfortable.


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