Please note that I'll be using the following notations as follows, throughout the question.
VX - For all X (Universal Quantifier)
EX - For some X(Existential quantifier)
Consider the logic formulas G1 and G2 :
G1 : (VX)[ P(X) V Q(X) ] --> (VX)P(X) V (VX)Q(X)
G2 : (EX)P(X) --> (VX)P(X)
Can anyone please explain how do we obtain the intuitive meaning of G1 and G2 under the interpretation I over the set of integers ; under which
P(X) means that X is even and
Q(X) means that X is odd.
To obtain the intuitive meanings of G1 and G2 under the given interpretation I over the set of integers, let's break down each formula:
G1: (VX)[ P(X) V Q(X) ] --> (VX)P(X) V (VX)Q(X)
To understand the intuitive meaning of this formula, we can analyze it clause by clause:
1. (VX)[P(X) V Q(X)]: This clause states that for all X in the set of integers, either P(X) or Q(X) is true. In our interpretation, P(X) means X is even and Q(X) means X is odd. Therefore, this clause can be interpreted as saying that for any integer X, it is either even or odd.
2. (VX)P(X) V (VX)Q(X): This clause states that there exists at least one even integer (P(X)) and there exists at least one odd integer (Q(X)). In other words, there are even numbers and there are odd numbers in the set of integers.
Combining both clauses, the formula states that in the set of integers, all numbers are either even or odd, and there exist even numbers as well as odd numbers. This aligns with our intuitive understanding of the set of integers.
G2: (EX)P(X) --> (VX)P(X)
Let's break down this formula:
1. (EX)P(X): This clause states that there exists at least one even number in the set of integers.
2. (VX)P(X): This clause states that for all X in the set of integers, X is even.
The formula can be interpreted as follows: If there is at least one even number in the set of integers, then all numbers in the set are even. This aligns with our understanding that if there is even one even number, all the numbers in the set of integers will be even.
Therefore, the intuitive meaning of G1 under the given interpretation is that all numbers in the set of integers are either even or odd, and G2 states that if there is at least one even number, then all numbers in the set are even.