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Every student in this class has studied calculus
If S(x) represents the statement that person x is in this class, C(x) which is the statement “x has studied calculus.”
we see that our statement can beexpressed as

x(S(x) C(x) )

Some student in this class has visited Mexico

M(x), which is the statement “x has visited Mexico.” S(x) to represent “x is a student in this class.”

x(S(x) M(x))

In 1st example Implication is taken  Why in 2nd Example and is taken

why not S(x) ->M(x)

For the second Statement  ∃x(S(x) M(x))  the student has to be in the class and has visited Mexico. Since ∃x is used, the result turns out to be true even if one student has visited Mexico. Suppose the student is not in the class then also the result turns out to be true if x (S(x) ->M(x)).

In first example if x is not student even then the result would be true

Can we replace implies with ^
Im unable to differentiate when to have implies , and in logic can u please help me

Think Logically by getting the basic definition of ∃ and ∀, ∃ is actually Infinite 'OR' i.e p1 or p2 or p3 or p4... and so on whereas ∀ is actually infinite 'AND' . Now if suppose your domain is students in your class and if you use ∀x(S(x) C(x) ) this makes sure that every student outside the domain is evaluated to True(you know about truth table of implication if the student is not in domain then s(x) is false and hence s(x)->c(x) is evaluated to True). Now suppose you take the same  ∃x(S(x) M(x)) if student is not in domain the result turns out to be true  and because of that the whole expression turns out to be true which we do not want we only want to check those students which are in the domain and hence x(S(x) ^ M(x))  is  the  right way according to the specification.

when domain is specified then we have to go for conjuction means and am i correct?

If you will replace ^ symbol with -> in 2nd example, you will found out like ∃x(S(x) -> M(x)) which means there exist a student,if student belong to the class then it must have visited Mexico, this is wrong actually , we should go like this, "There exist a student(some student /at least one or more) who belong to the class and he has visited Mexico" so it will go like this as x(S(x) ∧ M(x))

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we have mentioned their exist so for some x it is true

if x belongs to class and x visits mexico then result is true

if x belongs to class and x is not visiting mexico belongs false

why every x need to visit mexico
in second statement x refers to domain of only students it is true only for domain of students however for first domain is not specified so it is true for those who are not student of the class i hope this clearly demarks the difference between the two
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