Respuesta :
These are the preposition for the given conditions:
(P(1,1) ∧ P(1,2) ∧ P(1,3)) ∧ (P(2,1) ∧ P(2,2) ∧ P(2,3)) ∧ (P(3,1) ∧ P(3,2) ∧ P(3,3)).
(P(1,1) ∨ P(1,2) ∨ P(1,3)) ∨ (P(2,1) ∨ P(2,2) ∨ P(2,3)) ∨ (P(3,1) ∨ P(3,2) ∨ P(3,3)).
(P(1,1) ∨ P(1,2) ∨ P(1,3)) ∧ (P(2,1) ∨ P(2,2) ∨ P(2,3)) ∧ (P(3,1) ∨ P(3,2) ∨ P(3,3)).
(P(1,1) ∧ P(1,2) ∧ P(1,3)) v (P(2,1) ∧ P(2,2) ∧ P(2,3)) v (P(3,1) ∧ P(3,2) ∧ P(3,3)).
The domain of the propositional P(x, y) consists of pairs x and y, where x is 1, 2, 3 and y is 1, 2, 3.
What are the disjunction and conjunction?
In logic functions, there are two kinds of compound statements. These are referred to as conjunction and disjunction. A conjunction implies that both statements are correct, whereas a disjunction implies that at least one of the statements is correct. A conjunction connects statements with the word "and," whereas a disjunction connects statements with the word "or."
By using disjunction and conjunction we get,
a) ∀x∀y P(x, y) is written as:
(P(1,1) ∧ P(1,2) ∧ P(1,3)) ∧ (P(2,1) ∧ P(2,2) ∧ P(2,3)) ∧ (P(3,1) ∧ P(3,2) ∧ P(3,3)).
b) ∃x ∃y P (x, y) is written as:
(P(1,1) ∨ P(1,2) ∨ P(1,3)) ∨ (P(2,1) ∨ P(2,2) ∨ P(2,3)) ∨ (P(3,1) ∨ P(3,2) ∨ P(3,3)).
c) ∃x∀y P(x, y) is written as:
(P(1,1) ∨ P(1,2) ∨ P(1,3)) ∧ (P(2,1) ∨ P(2,2) ∨ P(2,3)) ∧ (P(3,1) ∨ P(3,2) ∨ P(3,3)).
d) ∀y∃x P(x, y) is written as:
(P(1,1) ∧ P(1,2) ∧ P(1,3)) v (P(2,1) ∧ P(2,2) ∧ P(2,3)) v (P(3,1) ∧ P(3,2) ∧ P(3,3)).
Hence, these are the preposition for the given conditions:
(P(1,1) ∧ P(1,2) ∧ P(1,3)) ∧ (P(2,1) ∧ P(2,2) ∧ P(2,3)) ∧ (P(3,1) ∧ P(3,2) ∧ P(3,3)).
(P(1,1) ∨ P(1,2) ∨ P(1,3)) ∨ (P(2,1) ∨ P(2,2) ∨ P(2,3)) ∨ (P(3,1) ∨ P(3,2) ∨ P(3,3)).
(P(1,1) ∨ P(1,2) ∨ P(1,3)) ∧ (P(2,1) ∨ P(2,2) ∨ P(2,3)) ∧ (P(3,1) ∨ P(3,2) ∨ P(3,3)).
(P(1,1) ∧ P(1,2) ∧ P(1,3)) v (P(2,1) ∧ P(2,2) ∧ P(2,3)) v (P(3,1) ∧ P(3,2) ∧ P(3,3)).
To learn more about the disjunction and conjunction visit,
https://brainly.com/question/1511319
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