So I have a couple questions on how to determine the elements in a set. I was hoping I could get an explanation on how to do these and if my answers were correct. Not really looking for answers just want to be pointed in the right direction.
a) $\{3^n | n ∈ N,n<10\}$
- $\{0,1,2,3,4,5,6,7,8,9\}$, is it supposed to be $3^n < 10$ or simply $n < 10$? I'm lost.
b) $\{4^n-72(2)^n + 512 = 0 | n ∈ N\}$
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$\{3,6\}$ I got this answer by substituting $x$ for $2^n$ which gave me:
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$x^2 – 72x + 512 = 0$ from there I got $x = 8$ or $x = 64$
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since $2^n = x$
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$2^n = 8$ or $2^n = 64$
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So, $n = 3$ or $6$
c) $\{n^3 – 4n^2 + 3n | n ∈ \{0,1,2,3,4\}\}$
- I'm really not sure what to do on this one. I think the subset is throwing me off. Should I factor $n$ out of the first half?
d) $A \times B$, where $A = \{n|n ∈ N, n/4\text{ is an integer}, n<20\}$
- no clue on this one.
Thanks in advance guys, really trying to wrap my head around this stuff
Best Answer
Your first question suggests you should start by reviewing set builder notation.
The lefthand portion of the notation specifies what the elements of the set actually are. For example, $\{3^n \mid n \in \mathbb{N}, n < 10\}$ says that the elements of the set are numbers of the form $3^n$.
The righthand portion of the notation further specifies any variables that might appear in the description of the elements. In our example, it says that the letter $n$ is supposed to be a natural number less than $10$.
Putting these two facts together, see if you agree with $$ \{3^n \mid n \in \mathbb{N}, n < 10\} = \{3^0, 3^1, 3^2, \dots, 3^9\} $$ (up to ambiguity about whether $0$ is a natural number or not).
See if this example helps you to approach the others. Ask questions in the comments if you need further clarification.