Rules for Using Predicates and Quantifiers
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In the realm of propositional logic, the study of logical equivalences involving quantifiers is a fascinating topic. These equivalences often help us simplify complex statements and make them easier to understand. However, a recent question has been raised: what happens if we swap the conjunction with disjunction in one equivalence, and disjunction with conjunction in another?
Let's take a look at the two important equivalences involving quantifiers:
- "There does not exist an x such that P(x) is true" is same as "For all x, P(x) is false."
- (\forall x (P(x) \wedge Q(x)) \equiv \forall x P(x) \wedge \forall x Q(x))
Now, let's see what happens when we replace the conjunction with disjunction in the first equivalence and the disjunction with conjunction in the second:
- "It is not true that for all x, P(x) is true" is not the same as "There exists an x such that P(x) is false."
- (\forall x (P(x) \vee Q(x)) \not\equiv \forall x P(x) \vee \forall x Q(x))
The latter is generally false because asserting that for all (x), either (P(x)) or (Q(x)) holds is not logically equivalent to saying that either (P(x)) holds for all (x) or (Q(x)) holds for all (x).
Similarly, the known equivalence (\exists x (P(x) \vee Q(x)) \equiv \exists x P(x) \vee \exists x Q(x)) does not hold if you replace the disjunction inside the quantifier with conjunction: (\exists x (P(x) \wedge Q(x)) \not\equiv \exists x P(x) \wedge \exists x Q(x)). This fails because (\exists x (P(x) \wedge Q(x))) requires there to be a single (x) that satisfies both (P) and (Q) simultaneously, while (\exists x P(x) \wedge \exists x Q(x)) only requires possibly different (x) values for (P) and (Q).
These results are supported by formal reasoning and explanations in the context of predicate logic equivalences[2]. Thus, only the original equivalences with conjunction for the universal quantifier and disjunction for the existential quantifier hold, and switching these connectives to their opposites invalidates the equivalences.
For those interested in testing their knowledge, practising questions from GATE CS Corner on Predicate Logic and quantifiers can help. Examples of such questions include questions from the years 2012, 2013, 2010, 2009, 2005, and 2016.
In conclusion, while logical equivalences involving quantifiers can be a powerful tool for simplifying complex statements, it's essential to remember that the connectives used within these equivalences matter. Swapping the conjunction with disjunction or vice versa can lead to incorrect results.
[2] Predicate Logic Equivalences, [Link to the reference]
In the field of education-and-self-development, solving mock tests focusing on predicate logic and quantifiers can help reinforce one's understanding. For instance, understanding the relationship between 'There exists an x such that P(x) is true' and 'It is not true that for all x, P(x) is true' requires learning about the use of the trie data structure, which mimics the structure of propositional logic formulas, in math and computer science.
