Can anyone help me in explaining this first order model with the following condition

Question

what is the explanation for not finding the following first order model such that model $N$ with domain $\{i,j,k,l\}$ and,

$N ⊨ ∀xB(h(x) ....(1)$

$N ⊨ ∃x¬B(x).... (2) $

while function h in this model $N$ is injunctive, please help me to explain why we can not have such model.

Answer 1

Hint

See Injective function: "every element of the function's codomain is the image of at most one element of its domain".

According to axiom (2), there is some element in $N = \{ i,j,k,l \}$ such that $B$ does not hold of it: assume for simplicity that $\lnot B(i)$.

According to axiom (1) we must have that:

$B(h(i)), B(h(j)), B(h(k)), B(h(l))$ all hold.

But, by injectivity, all of $h(i), h(j), h(k), h(l)$ must be distinct elemnts of $N$.