Quantifiers and Predicates - Set 2 Continuation
In the Graduate Aptitude Test in Engineering (GATE) Computer Science exam, questions on nested quantifiers have been a recurring theme, appearing in various years including 2010, 2009, 2005, 2013, 2016, and 2012. These questions, often found in the areas of Discrete Mathematics and Engineering Mathematics, test the candidate's understanding of predicate logic, particularly quantifiers.
The order of quantifiers is crucial in nested quantifiers, with a significant exception: when all the quantifiers are of the same kind, either all are universal or all are existential. For instance, the statement "There does not exist an x such that P(x) is true" is equivalent to "For all x, P(x) is false." Similarly, "It is not the case that every computer science graduate has taken a course in Discrete Mathematics" is equivalent to "There is a computer science graduate who has not taken a course in Discrete Mathematics."
However, it is essential to note that these logical equivalences do not remain valid if the conjunction is replaced with disjunction, or the disjunction is replaced with conjunction. For example, the statement "(\forall x (P(x) \lor Q(x)))" is not logically equivalent to "(\forall x P(x) \lor \forall x Q(x))". Similarly, "(\exists x (P(x) \land Q(x)))" is not logically equivalent to "(\exists x P(x) \land \exists x Q(x))".
These transformations do not preserve equivalence, a fact that can be understood intuitively, formally, and confirmed by the standard semantics of quantifiers and logical connectives. This is further confirmed by counterexamples showing that these alternate equivalences fail.
Here's a summary of the known equivalences and their replacement attempts:
| Original Equivalence | Replacement Attempt | Valid? | |------------------------------------------------------|---------------------------------------------|------------------------------| | (\forall x (P(x) \land Q(x)) \equiv \forall x P(x) \land \forall x Q(x)) | (\forall x (P(x) \lor Q(x)) \equiv \forall x P(x) \lor \forall x Q(x)) | No | | (\exists x (P(x) \lor Q(x)) \equiv \exists x P(x) \lor \exists x Q(x)) | (\exists x (P(x) \land Q(x)) \equiv \exists x P(x) \land \exists x Q(x)) | No |
In conclusion, the equivalences do not hold once the conjunctions and disjunctions are interchanged inside and outside the quantifiers.
Lastly, it is worth mentioning that a negated existential quantifier (¬∃x P(x)) turns into a universal quantifier (∀x ¬P(x)), and practicing questions on nested quantifiers can help test one's knowledge of the subject. Nested quantifiers are those where one quantifier is within the scope of the other, such as the interpretation of a statement with nested quantifiers: "For all x, there exists a y such that x + y = 0."
In the realm of education-and-self-development, mock tests can be beneficial for building understanding and confidence, especially in complex areas like Discrete Mathematics that are often featured in exams like the Graduate Aptitude Test in Engineering (GATE). Such mock tests might include questions involving nested quantifiers, like the question "For all x, there exists a y such that x + y = 0," which can help learners practice and apply their knowledge of quantifiers and logical connectives using a trie-like structure for effective learning.
