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?