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Semantics of First Order LogicConsider the universe U consisting of all finite lists of natural numbers, and a binary relation P that we interpret as the prefix-of ...

Question

Semantics of First Order LogicConsider the universe U consisting of all finite lists of natural numbers, and a binary relation P that we interpret as the prefix-of relation, that is P(T,y) holds if € is a prefix of y. For example_ we have that € is a prefix of the list [1,2,3] and only if x € {[L, [1], [1,2], [1,2,3]}. Formally; we are considering situation (U, 0) where U is the set of finite lists of natural numbers and O(P) {(I,y) prefix of y}: For each of the follwing formulae, determin

Semantics of First Order Logic Consider the universe U consisting of all finite lists of natural numbers, and a binary relation P that we interpret as the prefix-of relation, that is P(T,y) holds if € is a prefix of y. For example_ we have that € is a prefix of the list [1,2,3] and only if x € {[L, [1], [1,2], [1,2,3]}. Formally; we are considering situation (U, 0) where U is the set of finite lists of natural numbers and O(P) {(I,y) prefix of y}: For each of the follwing formulae, determine all situations (U, 0) where U is aS above, and 0( P) is the pretix relation, in which the respective formula is true, and justify your answer in a single sentence Vz Jygz(P(y;x) ^ P(z,1)) 4. Vy(Vz(P(T, 2) + P(z,y)) VzVy(P(z, y) V P(y; 1)) 5. Vx(P(T,y) - P(c,2)) VzVy(P(z,y) ~ P(z,2)) Vy( (Vz(P(E, 2)) - P(z;y))



Answers

A statement is in prenex normal form (PNF) if and only if it is of the form
$$
Q_{1} x_{1} Q_{2} x_{2} \cdots Q_{k} x_{k} P\left(x_{1}, x_{2}, \ldots, x_{k}\right)
$$
where each $Q_{i}, i=1,2, \ldots, k,$ is either the existential quantifier or the universal quantifier, and $P\left(x_{1}, \ldots, x_{k}\right)$ is a predicate involving no quantifiers. For example, $\exists x \forall y(P(x, y) \wedge Q(y))$ is in prenex normal form, whereas $\exists x P(x) \vee \forall x Q(x)$ is not (because the quantifiers do not all occur first).
Every statement formed from propositional variables, predicates, $\mathbf{T},$ and $\mathbf{F}$ using logical connectives and quantifiers is equivalent to a statement in prenex normal form. Exercise 51 asks for a proof of this fact.
Express the quantification $\exists 1 \times P(x),$ introduced in Section $1.4,$ using universal quantifications, existential quantifications, and logical operators.

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