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“For every person x, if person x is a student in this class then x has studied Calculus.”

S(x):Person x in the class

C(x):x has studied Calculus.

1.$\vartheta _{x} (S(x) --> C(x))$

2.$\vartheta _{x} (S(x) \Lambda C(x))$

Which one should hold? Why and why not?

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Best answer
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Name belongs to Class Calculus
abc  True True
def  True True
xyz  False True/False

I have constructed table on basis of given statement.

Consider first statement

$\forall _x (S(x) ->C(x))$

can be written as:

$(S(abc) -> C(abc)) \ \Lambda \  (S(def) -> C(def)) \ \Lambda \ (S(xyz) -> C(xyz))$

Substitute values, you will always get True.

Now,

$\forall _x (S(x)\  \Lambda \ C(x))$

can be written as:

$(S(abc)\  \Lambda \ C(abc)) \ \Lambda \  (S(def)\  \Lambda \ C(def)) \Lambda \ (S(xyz)\  \Lambda \ C(xyz))$

if you substitue values for $(S(xyz)\  \Lambda \ C(xyz)) = False \ \Lambda \ True/False = False$

And this will make entire expression false.

but the above expression should be true because xyz doesn't belong to class and we should not consider it, in implication $ False\ \Lambda \ True/False $  will be treated as True.

Hence only first statement is correct,

shortcut: $\forall$ is always followed by -> and $\exists$ is always followed by $\Lambda$.

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For every person x ,  if person x is a student in this class then x has studied Calculus

This must be true for every person , here the UoD is all the people in this universe.

Now if a person is a person belonging to the class , he must know Calculus , what if the person doesn't belong to the class and knows Calculus?

Here , "if person x is a student in this class then x has studied Calculus" is an implication statement , so even if some person doesn't belong to the class , he might or might not know calculus . 

What should the answer be then?

$\forall x(S(x) -> C(x))$.

What's the interpretation of this statement?

Let's take $x = \{"Ram","Shyam","Seeta","Geeta"\}$.

Ram and Shyam belong to the class and know Calculus , and Seeta and Geeta do not belong to the class but they also know Calculus.

Then,

$(S(Ram))->C(x)) \ and (S(Shyam)->C(Shyam)) \ and \ (S(Seeta)->C(Seeta)) \ and (S(Geeta)->C(Geeta))$

The above entire formula must be true , It will be false only in one case , if Somebody is a member of the class and doesn't know Calculus.

 

Why the other one is wrong?

$\forall x(S(x) and C(x))$. It would be true if and only if both S(x) and C(x) is true for every $x \in UoD$ .

This does not handle the case if $x$ doesn't belong to the class and knows Calculus.

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