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If I have for example the following:

$p$ is $ x > 4$

$\lnot p$ is $x < 4$ ?

direprobs
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1 Answers1

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For given real numbers $a$ and $b$, exactly one of the following holds: $$a<b$$ $$a=b$$ $$a>b$$

So the negation of any one of these is equivalent to the disjunction of the remaining two:

$$\lnot(a<b) \iff (a>b)\vee(a=b)$$ $$\lnot(a=b) \iff (a<b)\vee(a>b)$$ $$\lnot(a>b) \iff (a<b)\vee(a=b)$$

One usually sees "$a\geq b$" and "$a\leq b$" for the first and third cases. The second case is often written "$a\neq b$" for obvious reasons. Thus the symbol "$\geq$" is read "is greater than or equal to", etc.

MPW
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  • What do you mean by the "the negation of any one of these is equivalent to the disjunction of the remaining two"? – direprobs May 19 '16 at 18:24
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    @direprobs That means that if one of the statements is false, then one or the other of the remaining two must be true. ("Disjunction" is the "or" relation.) – Théophile May 19 '16 at 18:58