Please explain in a different way


Define a set X of numbers as follows
 B. 2 belongs X.
R1. If x belongs to X, so is 10x
R2. If x belongs X, so is x+4

a. List all the elements of X that are less than 30.
b. Explain why there are no odd numbers in X.

2 belongs to X
x belongs to X ==> 10x belongs to X
x belongs to X==>x+4 belongs to x

Find all of X belongs to 30
2            2     20
20          6     24
6           10    28
24         14

odds not belongs to x
4 not belongs to x
8 not belongs to x
12 not belongs to x
16 not belongs to x
Who is Participating?
Okay. So B is base and R is rule. I get the whole idea now. Sorry about the other approach in the last question. We did recursive sets a different way.

So you know that 2 is in X and for every element x that is in X 10x and 4+x are also in X.

So 2 is in X because of B
20 is in X because of R1 (2 is in X so 10*(2) is in X)
6, 10, 14, 18, 22, 26 are all in X from R2 (2 is in X 2+4 = 6. 6+4 = 10 etc, so all are in X)

Also, since 20 is in X, 24 and 28 also are.

There are no odd numbers in X since 10 and 4 are even numbers so any combination of multiplying by 10 and adding 4 will still be even.

Or, more formally, if x is an even number, then 10x is even
If x is an even number, x+4 is even.
2 is even.
So all numbers in X are even.
Once you know that 18 and 20 are in X, then all even numbers greater than 20 are also in X since you can get to any even greater than 20 by starting with either 18 or 20 and adding 4 until you get there.
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