D421 Discrete Math: Functions and Relations - Set 2 - Part 3

Correct Answer: They have identical truth values in every case

Two propositions are logically equivalent when they have the same truth value for every possible assignment of their variables.

Correct Answer: p ∧ (q ∨ r) equals (p ∧ q) ∨ (p ∧ r)

The distributive law states p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r), mirroring distribution in algebra.

Correct Answer: p ∧ true equals p

The identity law states that conjoining a proposition with true leaves it unchanged: p ∧ T = p.

Correct Answer: p ∨ true equals true

The domination law states p ∨ T = T, since a disjunction with true is always true regardless of p.

Correct Answer: For every x there exists a y making P(x, y) true

∀x ∃y P(x, y) means for each x, some y (possibly depending on x) makes the predicate true.

Correct Answer: No, the order of quantifiers matters

The order of mixed quantifiers matters: ∃y ∀x requires a single y working for all x, a stronger claim than ∀x ∃y.

Correct Answer: A sequence of premises followed by a conclusion

An argument consists of premises and a conclusion. It is valid if the conclusion follows necessarily from the premises.

Correct Answer: When the conclusion is true whenever all premises are true

An argument is valid when it is impossible for all premises to be true while the conclusion is false, regardless of actual truth.

Correct Answer: Sound means valid and all premises are actually true

A valid argument has correct logical form; a sound argument is valid and additionally has all true premises, guaranteeing a true conclusion.

Correct Answer: To rigorously establish the truth of statements through valid reasoning

Formal logic provides the rigorous framework of definitions, inference rules, and proofs used to establish mathematical truth with certainty.