Can we apply the CRT to this system: \[\begin{align*} x \equiv_{10} 3 \\ x \equiv_{2} 1 \\ x \equiv_3 2 \end{align*}\]
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Can we apply the CRT to this system: \[\begin{align*} x \equiv_{10} 3 \\ x \equiv_{2} 1 \\ x \equiv_3 2 \end{align*}\]
yes we can, even though \(\gcd(10, 2) = 2\), as we can decompose \(x \equiv_{10} 3\) into \(x \equiv_5 3\) and \(x \equiv_2 3 \equiv_2 1\) which matches the other equation. Thus the solution is still unique.
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | Can we apply the CRT to this system: \[\begin{align*} x \equiv_{10} 3 \\ x \equiv_{2} 1 \\ x \equiv_3 2 \end{align*}\] | |
| Back | yes we can, even though \(\gcd(10, 2) = 2\), as we can decompose \(x \equiv_{10} 3\) into \(x \equiv_5 3\) and \(x \equiv_2 3 \equiv_2 1\) which matches the other equation. Thus the solution is still unique. |
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What is really important for the prenex form due to the binding of quantifiers?
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What is really important for the prenex form due to the binding of quantifiers?
We need to wrap the entire expression in parentheses \(\forall \exists (...)\) otherwise, it's not prenex!
Field-by-field Comparison
| Field | Before | After |
|---|---|---|
| Front | What is really important for the prenex form due to the binding of quantifiers? | |
| Back | We need to wrap the entire expression in parentheses \(\forall \exists (...)\) otherwise, it's not prenex! |