D420 Discrete Math: Functions and Relations - Set 4 - Part 3

Correct Answer: A set of rules that describe how negations interact with conjunctions and disjunctions

De Morgan's Law describes how negations distribute over conjunctions and disjunctions, specifically: (pq) = pq and (pq) = pq.

Correct Answer: It asserts that there exists at least one element in the domain for which the predicate is true

The existential quantifier x asserts that there is at least one element in the domain for which the predicate P(x) is true.

Correct Answer: The predicate is true for all elements in the domain

The universal quantifier x asserts that the predicate P(x) is true for every element in the domain.

Correct Answer: Assuming the negation of the statement to be proved and deriving a contradiction

A proof by contradiction assumes the negation of the statement to be proved and then shows that this assumption leads to a contradiction, thereby proving the original statement.

Correct Answer: Proving a statement by directly showing that the hypothesis leads to the conclusion

A direct proof involves starting with the hypothesis and making logical steps that lead to the conclusion.

Correct Answer: To prove a statement by proving its contrapositive, qp

A proof by contrapositive involves proving the contrapositive of a statement, which is logically equivalent to proving the original statement.

Correct Answer: A proof that verifies a statement by checking all possible cases

A proof by exhaustion involves checking all possible cases to verify that the statement holds for each one.

Correct Answer: A proof that involves a base case and an inductive step

A proof by mathematical induction involves proving a base case and then showing that if the statement holds for one case, it holds for the next case.

Correct Answer: The initial case for which the statement is proven to be true

The base case in mathematical induction is the initial case where the statement is directly proven to be true, forming the foundation for the inductive step.

Correct Answer: The inductive step assumes the statement is true for all previous cases up to the current one

In strong mathematical induction, the inductive step assumes that the statement is true for all previous cases up to the current one, rather than just for one prior case.