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So the following equation's set notation would be:

w = { t ∈ ℤ  │ -1 < t < 3 }

w = {-1, 0, 1, 2, 3}

correct?

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Now, what happens if I move the negative sign to 3, does my set notation flip, or is it now considered void?

w = { t ∈ ℤ  │ 1 < t < -3 }

w = {1, 0, -1, -2, -3}

or w = { ∅ } ?
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In the first case w = {0, 1, 2}, not {-1, 0, 1, 2, 3}. -1 < -1 and 3 < 3 are both false statements.

None of the numbers in your second set, {1, 0, -1, -2, -3}, are greater than 1, so none of them should be in the set w.

w = ∅, not {∅}. They're not the same thing.

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