Final answer:
Demorgan's Laws are a set of fundamental rules in logic that simplify expressions involving negations of conjunctions and disjunctions. The distributive laws for sets describe how to combine unions and intersections of sets.
Step-by-step explanation:
Demorgan's Laws:
Demorgan's Laws are a set of two fundamental rules in logic that describe how to simplify expressions involving negations of conjunctions and disjunctions. The laws state that the negation of a conjunction (AND) is equivalent to the disjunction (OR) of the negations of the individual terms, and the negation of a disjunction (OR) is equivalent to the conjunction (AND) of the negations of the individual terms.
For example, if we have the expression ¬(A ∧ B), according to Demorgan's Laws, it can be simplified to (¬A ∨ ¬B). Similarly, the expression ¬(A ∨ B) can be simplified to (¬A ∧ ¬B).
Distributive Laws for Sets:
The distributive laws for sets describe how to combine unions and intersections of sets. The laws state that the union of two sets intersected with a third set is equivalent to the intersection of the unions of the two sets and the third set. Similarly, the intersection of two sets unioned with a third set is equivalent to the union of the intersections of the two sets and the third set.
For example, let's say we have three sets A, B, and C. According to the distributive laws, (A ∪ B) ∩ C is equivalent to (A ∩ C) ∪ (B ∩ C). Similarly, (A ∩ B) ∪ C is equivalent to (A ∪ C) ∩ (B ∪ C).