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Express the statement “Every student in this class has studied calculus” using predicates and
quantifiers.
Solution: First, we rewrite the statement so that we can clearly identify the appropriate quantifiers
to use. Doing so, we obtain:
“For every student in this class, that student has studied calculus.”
Next, we introduce a variable x so that our statement becomes
“For every student x in this class, x has studied calculus.”
Continuing, we introduce C(x), which is the statement “x has studied calculus.” Consequently,
if the domain for x consists of the students in the class, we can translate our statement as ∀xC(x).
However, there are other correct approaches; different domains of discourse and other
predicates can be used. The approach we select depends on the subsequent reasoning we want
to carry out. For example, we may be interested in a wider group of people than only those in
this class. If we change the domain to consist of all people, we will need to express our statement
as
“For every person x, if person x is a student in this class then x has studied calculus.”
If S(x) represents the statement that person x is in this class, we see that our statement can be
expressed as ∀x(S(x) → C(x)). [Caution! Our statement cannot be expressed as ∀x(S(x) ∧
C(x)) because this statement says that all people are students in this class and have studied
calculus!]
Finally, when we are interested in the background of people in subjects besides calculus,
we may prefer to use the two-variable quantifier Q(x, y) for the statement “student x has
studied subject y.” Then we would replace C(x) by Q(x, calculus) in both approaches to obtain
∀xQ(x, calculus) or ∀x(S(x) → Q(x, calculus)).

I din't get the last paragraph ,
it says "when we are interested in the background of people in subjects besides calculus"
So, a two-variable quantifier Q(x, y) is used then  replace C(x) by Q(x, calculus)
Doubt 1: What is x here? Is it a subject besides calculus that must be known to every people who are reading in the class.
Doubt 2: Why the value calculus is given as input now? In 2nd last paragraph “For every person x, if person x is a student in this class then x has studied calculus.” gives, ∀x(S(x) → C(x)), here its C(x) not C(calculus). So, the value is not given as input there. Then why now?

edited | 171 views
+1

First read it more than 5 times, you may get that

Otherwise read my approach..

given that " Every student in this class has studied calculus " ------> Question

given that C(x) ----- X studied Calculus ( where x is a person and Calculus is a subject )

if domain of person is Student then it can be represented as ∀xC(x).

if domain of person is not only Student then it can be represented as ∀x ( S(x) → C(x) ). means if a person is a student then he studied Calculus

now they want to change the domain of subject

given that  Q(x, y) for the statement “ x has studied subject y”  ( where x is a person and y is a subject )

if your subject is Calculus then Q(x,y) changed as Q(x,calculus).

if domain of person is Student then it can be represented as ∀xQ(x,calculus).

if domain of person is not only Student then it can be represented as ∀x ( S(x) → Q(x,calculus) ). means if a person is a student then he studied Calculus

Actually given that  Q(x, y) for the statement “ student x has studied subject y ”

but i changed it as Q(x, y) for the statement “  x has studied subject y ”  for the purpose of changing the domain of x

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Thanks.
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@Shaikh sir

If C(x)= x is a student in class

S(x)= x studied math

For the stmt represent using predicate calculus

1)Every student in class studied math

∀(C(x)→S(x))

2)Some student in class studied math

∃(C(x)⋀S(x))

My doubt is why can't we use

For 1  stmt ∀(C(x)⋀S(x))

And

For 2 stmt ∃(C(x)→S(x))
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∀(P(x))  expands to suppose P1⋀P2^P3^....P10, where P1 to P10 is the domain.
In the case of ∀(C(x)⋀S(x)), if the person P2  doesn't belong to the class then ∀(C(x)⋀S(x)) will be false.;
But it should be true because we don't care about the person who doesn't come to class.

In case of   ∀(C(x)→S(x)) , if C(x) is false(person who doesn't come to class) then its true . and if C(x) is false it depends on if S(x) is true or false.

So, if a person who doesn't come to class it's always true,

Similarly in ∃(P(x)), expands to suppose P1 or P2 or P3 or ....P10, where P1 to P10 is the domain.
In case of  ∃(C(x)→S(x)) if a person who doesn't come to class it's always true,
But it should be false because we don't care about the person who doesn't come to class.
So,  we use ∃(C(x)⋀S(x))
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@sandy good explanation....

@Rohit, if you didn't get this... Read the caution line in question... Actually it's not a question, it's a description of concept

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