Confined Quantifiers - Re Expressing formula

Question

so I'm doing some study on Confined Quantification, and I understand how it when converting to english, but I don't understand how to re express the formula? I've tried watching videos but nothing has really helped me. The practice question is:

Re-express the following formulas using confined quantifiers, where the confining set is an integer interval.

(a) X =A1 ∨X =A2 ∨...∨X =An

(b) A1 ̸=A2 ∧A2 ̸=A3 ∧...∧An−1 ̸=An

Can anyone provide me an explanation on how to complete these? Much appreciated Thanks.

Answer 1

Assuming that "Confined Quantification" is the same as restricted (or guarded) quantification, in order to :

Re-express the following formulas using confined quantifiers, where the confining set is an integer interval :

(a) $X = A_1 \lor X = A_2 \lor...\lor X = A_n$,

we must intorudece the set $N = \{1, 2, ... n \}$, we may re-express it as :

$(\exists i \in N)(X = A_i)$.

For

(b) $A_1 \ne A_2 \land A_2 \ne A_3 \land... \land A_{n−1} \ne A_n$

we need $N' = \{1, 2, ... n-1 \}$:

$(\forall i \in N')(A_i \ne A_{i+1})$.