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# Number Sets

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Let $$A = \left \{ x ∈N | x^2 < 37 \right \}$$ and $$B = \left \{ 3k + 1 |k ∈ \left \{ 1, 2, 3, 4 \right \} \right \}$$ be sets.

1. List the elements of $$A$$ and $$B$$, and the subsets of $$B$$.

2. For any sets $$A$$$$B$$, and $$C$$, prove or disprove the claim: If $$A ∈ B$$$$B ∈ C$$, then $$A ∈ C$$.

3. Show $$\left \{ 12a + 4b |a, b ∈ Z \right \} = \left \{ 4c |c ∈Z \right \}$$

Aug 23, 2022

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I assume $$N$$ is for natural numbers and $$Z$$ is for integers.

1.

A = {1, 2, 3, 4, 5, 6}

B = {4, 7, 10, 13}

Subsets of B are any combination of the elements of B (excluding some).

2.

If A is contained within B, and B is contained with C, A is contained within C.

This is, I suppose, one of those annoying proofs which I take for granted and see no way of prooving it other than it is logical. It is here I should also note that I have never studied set theory, and I'm probably getting everything wrong .

3.

{4(3a+b)|a,b $$\in$$Z} = {4c|c$$\in$$Z}

{3a+b|a,b$$\in$$Z}={c|c$$\in$$Z}

3a+b$$\in$$Z

c$$\in$$Z

(I don't know what to do after that....)

Again, I've never studied this, so apologies in advance for faulty logic everywhere.

Aug 23, 2022