Using truth tree decompostition, determine the consistency of the set of proposittions. Include line numbers and justifications, and recover the truth values if consistent.

{A and B, ~B or C}

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To determine the consistency of the set of propositions {A and B, ~B or C} using truth tree decomposition, we need to construct a truth tree and check if there is at least one branch with all open propositional variables (atoms) assigned to consistent truth values.

First, let's list the given propositions with line numbers:

1. A and B
2. ~B or C

To begin constructing the truth tree, we will consider the negation of the conjunction in proposition 1 and the disjunction in proposition 2. This allows us to apply the appropriate rules of inference.

Step 1: Negate the conjunction in proposition 1:
1. A and B ------------ (Given)
|- ~(A and B) ------ (Negation)

Step 2: Negate the disjunction in proposition 2:
1. A and B ------------ (Given)
2. ~B or C ------------ (Given)
|- ~(~B or C) ------- (Negation)

Now we have established the initial branch. We will branch off twice based on the two negated propositions in step 2.

Branch 1: ~(A and B)
Branch 2: ~(~B or C)

Let's analyze each branch.

Branch 1: ~(A and B)
1. A and B --------- (Given)
1. ~(A and B) ------- (Assumption)
|- A ------------ (Disjunction Elimination)
|- B ------------ (Disjunction Elimination)
|- ~A ------------ (Negation Introduction)
|- ~B ------------ (Negation Introduction)

From this branch, we can see that both A and B, as well as their negations ~A and ~B, appear. This branch is inconsistent, as it contains both a proposition and its negation.

Branch 2: ~(~B or C)
1. A and B --------------- (Given)
2. ~B or C --------------- (Given)
2. ~(~B or C) ----------- (Assumption)
|- B ----------------- (Disjunction Elimination)
|- C ----------------- (Disjunction Elimination)
|- ~B ---------------- (Negation Introduction)
|- ~C ---------------- (Negation Introduction)

From this branch, we can see that both B and ~B appear. This branch is also inconsistent.

Since both branches yielded inconsistencies, it means that the given set of propositions {A and B, ~B or C} is inconsistent. There is no branch where all open propositional variables have consistent truth values.

Hence, there is no consistent truth assignment that satisfies both propositions A and B and ~B or C.