Which of the following options are corrent in propositional logic and why?

Question

A, B and C are statements in propositional logic which of following is correct and why? (the corrent answer is 3 but why) ($⊨$ is entailment)

1. $[A ∧ B ⊨ C] ⇒ [A ⊨ C] ∨ [B ⊨ C]$
2. $[A ∨ B ⊨ C] ⇒ [A ⊨ C] ∧ [B ⊨ C]$
3. $[A ⊨ C] ∨ [B ⊨ C] ⇒ [A ∧ B ⊨ C]$
4. $[A ⊨ C] ∨ [B ⊨ C] ⇒ [A ∨ B ⊨ C]$

why 1 is correct:

$A ⊨ B ≡ A ⇒ B ≡ ¬A ∨ B$

so

$(¬A ∨ ¬B) ∨ C ≡ (¬A ∨ ¬B) ∨ C ∨ C ≡ (¬A ∨ C) ∨ (¬B ∨ C) ≡ (A ⇒ C) ∨ (B ⇒ C) ≡ (A ⊨ C) ∨ (B ⊨ C)$

why 2 is correct:

$(¬A ∧ ¬B) ∨ C ≡ (¬A ∨ C) ∧ (¬B ∨ C) ≡ (A ⇒ C) ∧ (B ⇒ C) ≡ (A ⊨ C) ∧ (B ⊨ C)$

why 3 is correct

$(¬A ∨ C) ∨ (¬B ∨ C) ≡ (¬B ∨ ¬A) ∨ C ≡ ¬(B ∧ A) ∨ C ≡ (B ∧ A) ⇒ C ≡ A ∧ B ⊨ C$

Answer 1

$A\models B \equiv A\to B$ is not correct.   The left hand statement is that $A$ semantically entails $B$ -- that the model has no interpretation where $A$ is satisfied but $B$ not.   The right hand statement is simply that $A$ materially implies $B$ -- that the current evaluations for $A,B$ are such that the implication is valued as true.

From $A\wedge B\models C$ we cannot make the inference that $A\models C$ or $B\models C$.   $A\wedge B\models C$ allows an interpretation where $A$ is true, $B$ is false, and $C$ is false.   It also allow an interpretation where $A$ is false, $B$ is true, and $C$ is false.   Then we may have a model where $A\wedge B\models C$ holds but both $A\models C$ and $B\models C$ fail.

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It is somewhat analogous to the difference between $\forall x~ (A{\small(}x{\small)}\wedge B{\small(}x{\small)} \to C{\small(}x{\small)})$ and $A{\small(}e{\small)}\wedge B{\small(}e{\small)}\to C{\small(}e{\small)}$, and how $\forall x~((A{\small(}x{\small)}\wedge B{\small(}x{\small)})\to C{\small(}x{\small)})$ is not equivalent to $(\forall x~(A{\small(}x{\small)}\to C{\small(}x{\small)}))\vee(\forall x~(B{\small(}x{\small)}\to C{\small(}x{\small)})$,

Answer 2

To see that statement 1 is incorrect, consider what happens when $A$, $B$, and $C$ are $p\to q$, $p$, and $q$, respectively, where $p$ and $q$ are propositional variables. Then $A\land B\models C$ says that $q$ is a logical consequence of $(p\to q)\land p$, which is correct. But $A\models C$ says that $q$ is a logical consequence of $p\to q$, which is not correct, as $p$ and $q$ might both be false, which would make $p\to q$ true. And $B\models C$ says that $q$ is a logical consequence of $p$, which is not correct either, as $p$ might be true while $q$ is false.

Similarly, to see that statement 4 is incorrect, consider the case where $A$ and $C$ are both the propositional variable $p$ while $B$ is a different propositional variable $q$.

On the other hand, statements 2 and 3 both look correct to me.