Suppose the domain of the propositional function P(x,y) consists of pairs x and y, where x = 1, 2, or 3, and y = 1, 2, or 3. Write out the following proposition using disjunctions and conjunctions only:
∃xP(x,y)
I don't even know where to start... (x ∧ y) ∨ (y ∧ x) is an ok answer? I don't even know how that would make sense anyway though, so any help would be greatly appreciated.
To write out the proposition ∃xP(x,y) using only disjunctions and conjunctions, we need to express the existence of an x such that P(x,y) is true.
Here's the step-by-step process to tackle this:
1. Start with the domain of the propositional function P(x, y), which consists of pairs x and y, where x = 1, 2, or 3, and y = 1, 2, or 3. Now we have nine possible pairs: (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), and (3, 3).
2. To express the existence of an x such that P(x, y) is true, we need to evaluate all the combinations of P(x, y) and use conjunctions and disjunctions to connect them.
3. For each pair (x, y), determine whether P(x, y) is true or false. Let's say P(x, y) is true for the pairs (1, 2), (2, 3), and (3, 1), but false for all other pairs.
4. Now, we connect the propositions P(x, y) using disjunctions (∨). We need to create a disjunction (∨) with all the true propositions. In our case, the true propositions are P(1, 2), P(2, 3), and P(3, 1). So, the disjunction would be (P(1, 2) ∨ P(2, 3) ∨ P(3, 1)).
5. Finally, the complete proposition ∃xP(x,y) would be formed by wrapping the disjunction in an existential quantifier (∃x). So, the final proposition using disjunctions and conjunctions only would be ∃x(P(x, y)) = ∃x(P(1, 2) ∨ P(2, 3) ∨ P(3, 1)).
Note: The answer (x ∧ y) ∨ (y ∧ x) is not correct for this particular proposition. That expression would represent a combination of x and y, rather than expressing the existence of an x for which P(x, y) is true.