Negation of propostitions with quantifiers

[razored]

I need a little help deciphering my text. It says as follows :
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a) Some men are soldiers.
b) All men are hungry.

More formally these are stated :
a)$$\exists_{x} p$$, where x belongs to the set of men and p is the proposition : x is a soldier.
b)$$\forall_{x}$$,where x belongs to the set of men and q is the proposition : x is hungry.

The correct negations of the above propositions are:
a)All men are not soldiers, or: $$\forall_{x}[$$~$$p]$$
b)Some men are not hungry, or: $$\exists_{x}[$$~$$p]$$

You should examine carefully the reasons for rejecting the following statements as suitable negations.

a)Some men are not soldiers.
b)All men are not hungry.

Remember that the negation of a true proposition must be false.
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They say that the negation of a true proposition must be false. Like " All men are hungry.(true)" then, "All men are not hungry.(false)" Why can't we use that? Its negation appears to be false. Can anyone give me a better explanation? Also, would a truth table help me here?

 

[HallsofIvy]

I need a little help deciphering my text. It says as follows :
------------------------------------------------------------------------------------------
a) Some men are soldiers.
b) All men are hungry.

More formally these are stated :
a)$$\exists_{x} p$$, where x belongs to the set of men and p is the proposition : x is a soldier.
b)$$\forall_{x}$$,where x belongs to the set of men and q is the proposition : x is hungry.

The correct negations of the above propositions are:
a)All men are not soldiers, or: $$\forall_{x}[$$~$$p]$$
b)Some men are not hungry, or: $$\exists_{x}[$$~$$p]$$

You should examine carefully the reasons for rejecting the following statements as suitable negations.

a)Some men are not soldiers.

It is quite possible that some men are soldiers and some men are NOT soldiers. Those can both be true.

b)All men are not hungry.

Saying "It is not true that all men are hungry" does NOT mean that NO men are hungry.

Remember that the negation of a true proposition must be false.
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They say that the negation of a true proposition must be false. Like " All men are hungry.(true)" then, "All men are not hungry.(false)" Why can't we use that? Its negation appears to be false. Can anyone give me a better explanation? Also, would a truth table help me here?

Yes, it is true that the negation of a true proposition must be false. It does NOT follow that ANY false statement is the negation of a true proposition!

 

[razored]

"Saying "It is not true that all men are hungry" does NOT mean that NO men are hungry."
This means that some men are hungry and some aren't?

Also, I need help deciphering this phrase :
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$$\forall_{x1} \forall_{x2} [$$ If $$x_{1}$$ is congruent to $$x_{2}$$, the median of $$x_{1}$$ equals the median of $$x_{2}]$$

Negation :
$$\exists_{x1} \exists_{x2} [ x_{1}$$ and $$x_{2}$$ are congruent, and the median of $$x_{1}$$ does not equals the median of $$x_{2}$$]

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In the first statement, does it mean for ALL triangles X1 and X2, if they are congruent then the medians are equal ?

The second statement says for SOME triangles X1 and X2, if they are congruent, then the medians don't equal?

Thanks.

 

[HallsofIvy]

"Saying "It is not true that all men are hungry" does NOT mean that NO men are hungry."
This means that some men are hungry and some aren't?

Also, I need help deciphering this phrase :
--------------------------------------------------
$$\forall_{x1} \forall_{x2} [$$ If $$x_{1}$$ is congruent to $$x_{2}$$, the median of $$x_{1}$$ equals the median of $$x_{2}]$$

Negation :
$$\exists_{x1} \exists_{x2} [ x_{1}$$ and $$x_{2}$$ are congruent, and the median of $$x_{1}$$ does not equals the median of $$x_{2}$$]

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In the first statement, does it mean for ALL triangles X1 and X2, if they are congruent then the medians are equal ?

Assuming we are given that X1 and X2 are triangles, yes, that is what it says.

The second statement says for SOME triangles X1 and X2, if they are congruent, then the medians don't equal?

It might be better to read it as "there exist at least one pair of triangles that are congruent but their medians are not equal" (of course, since the first statement is true, that statement is false).

Thanks.