Given: \[A = \{x: 1 \le x \le 5\}\] why is \[A' = \{x : x < 1 \vee x > 5\}\] and not \[A' = \{x: x< 1 \wedge x > 5\}\]

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Given: \[A = \{x: 1 \le x \le 5\}\] why is \[A' = \{x : x < 1 \vee x > 5\}\] and not \[A' = \{x: x< 1 \wedge x > 5\}\]

Mathematics
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Since \(x\) cannot be both less than one and greater than five at one time (otherwise we have a logical contradiction). Also, De Morgan's laws are useful for this.
so does that mean \[A = \{x : x \ge 1 \wedge x \le 5\}\]
In any case: \[ A=\{x|1\leq x\leq 5\}=\{x|1\leq x \wedge x\leq 5\} \]

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Other answers:

Yessir.
i can't believe i didn't think of that first reason....
Unless your x's can quantum tunnel . . .
uh-oh....math conversation.....time to leave..
(Sorry, I've been hanging out in the Physics section too long)
Go go, quantum entanglement and random transmission of data!
Strange things can happen in this here multiverse.

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