In propositional logic, a formula in Conjunctive Normal Form (CNF) is termed as "glimsy" if for every integer j ≥ 1, any group of j conjuncts contains a propositional variable that is seen only once among those conjuncts. When counting occurrences of propositional variables among a group of conjuncts, the presence of the variable in its negated or non-negated state is immaterial; both are considered. For illustration: Consider the formula (p ∨ q) ∧ (q ∨ ¬r) ∧ (r ∨ ¬s). This is termed as glimsy. On analyzing, if you pick j = 1, each conjunct has a propositional variable that is singular in its occurrence (in actuality, every variable in the conjunct is unique). The repetition in other conjuncts is irrelevant for a specific conjunct's evaluation. For j = 2, choosing conjuncts p ∨ q and q ∨ ¬r, p and r are singular in their occurrences. This holds true for other combinations of two conjuncts as well. Lastly, for j = 3, including all the three conjuncts, p and s occur only once, making the condition true. However, the formula (p ∨ q) ∧ (q ∨ ¬r) ∧ (¬p ∨ ¬r) ∧ (p ∨ ¬s) isn't glimsy. Although some groups of conjuncts meet the criteria, the first three conjuncts together contain p, q, and r twice, and no other variable, violating the glimsy definition. Can we prove by induction that a glimsy formula in CNF using a maximum of m variables comprises at most m conjuncts and is valid?
In propositional logic, a formula in Conjunctive Normal Form (CNF) is termed as "glimsy" if for every integer j ≥ 1, any group of j conjuncts contains a propositional variable that is seen only once among those conjuncts. When counting occurrences of propositional variables among a group of conjuncts, the presence of the variable in its negated or non-negated state is immaterial; both are considered.
For illustration:
Consider the formula (p ∨ q) ∧ (q ∨ ¬r) ∧ (r ∨ ¬s). This is termed as glimsy.
On analyzing, if you pick j = 1, each conjunct has a propositional variable that is singular in its occurrence (in actuality, every variable in the conjunct is unique). The repetition in other conjuncts is irrelevant for a specific conjunct's evaluation. For j = 2, choosing conjuncts p ∨ q and q ∨ ¬r, p and r are singular in their occurrences. This holds true for other combinations of two conjuncts as well. Lastly, for j = 3, including all the three conjuncts, p and s occur only once, making the condition true.
However, the formula (p ∨ q) ∧ (q ∨ ¬r) ∧ (¬p ∨ ¬r) ∧ (p ∨ ¬s) isn't glimsy.
Although some groups of conjuncts meet the criteria, the first three conjuncts together contain p, q, and r twice, and no other variable, violating the glimsy definition.
Can we prove by induction that a glimsy formula in CNF using a maximum of m variables comprises at most m conjuncts and is valid?
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This proof is incorrect, as part of the quesiton, it is possible.
Look at the first example:
(p ∨ q) ∧ (q ∨ ¬r) ∧ (r ∨ ¬s).
Both p & ¬s are unique, and satisfy glimsy.
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