Kenneth Rosen Edition 6th Exercise 1.3 Example 17 (Page No. 38)

Question

The restriction of a universal quantification is the same as the universal quantification
of a conditional statement. For instance, ∀x < 0 (x2 > 0) is another way of expressing
∀x(x < 0 → x2> 0). On the other hand, the restriction of an existential quantification is the
same as the existential quantification of a conjunction. For instance, ∃z > 0 (z2 = 2) is another
way of expressing ∃z(z > 0 ∧ z2 = 2).

Ques : Why universal quantification is same as universal quantification of a conditional statement whereas existential quantification is same as existential quantification of a conjunction?
 

Please provide proper details. Thank You.
 

Answer 1

universal quantification of a conditional statement can also be explained as like this,

∀x(x < 0 → x²> 0)= let x₁,x₂,x_3,....  are instances of x, according to definition of universal quantification, we can elaborate this as,

x₁<0, x₁² >0 and x₂<0, x^₂2 >0 and x₃<0, x₃² >0 ..........

if any statement is false, the whole equation is false, so all statement must be true, for final equation to  be true,

in case of universal quantification of a conditional statement we apply case of less than 0 to every statement when it is processed,

in case of universal quantification,∀x < 0 (x² > 0)  we initially take out x which match condition less then 0, then make statement x² > 0,

as you said these both mean same, but representation are different,

same is the case with existential quantification