How to represent the below proposition using a well-defined formula ?

Question

The only even prime is 2.

Answer 1

Let x = set of natural numbers.

EQUAL(X,Y) : 'X' is equal to 'Y'

EVEN(X) : 'X' is even.

PRIME(X) : 'X' is prime.

"The only even prime is "2"  can be expressed as :

∀x [ (EVEN(X) ∧ PRIME(X)) ↔ EQUAL(X,2)]

Here bi-direction is necessary because if only uni-direction implication were there from Left to Right and consider if number is even but prime ,so LHS becomes false so we can't say anything about RHS i.e whether number is two or not.This is so because in implication,if LHS is false we can't say anything about RHS and whole implication becomes true i.e F -> T or F is always true.

Hence we need both side implication to show that if given no. is both prime and even then it is only '2'  ,and if given no. is '2' then its even & prime both.

I hope it clear many doubts related to implication.

Answer 2

I am not sure but you can try my option and let me know if i m wrong....

See Question is saying No. Should be even as well as prime(even and prime) and if it is true then no.has to be 2 otherwise formula sholud not exist....

If x={set of all integers} and

P(x):No.is even

Q(x):No.is Prime

R(x):No. is 2

then

for all x (P(x)^ Q(x)<--->R(x))

Double Implication bcz if no.is even and prime then it has to be 2 otherwise both has to be false

Comments

what about the case when LHS is false while RHS is true ,I guess this will never happen since if R(x) is true that implies 2 is even which is true so then what is the significance of implication here ?

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yes, double implication should be there..

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now sirjiii??

 

On that day we didn't had much discussion on double implication but now it is clear thanku :)

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why u chose the domain as "for all " ,since we have only 1 choice, so why not use unique quanitifier .

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unique is not a standard quantifier in first order logic..

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Why double implication shd be der...plz elaborate..

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Because the given statement says "only even prime is 2". It means

"If any number is even and prime it is 2" one side of implication.

It also means, "2 is even and is prime" other side of implication.

Answer 3

∃x=2 A(x)

where A(x): x is even and prime

domain of x: set of all natural numbers

we can express this  statement with existential  quantifier  .