Answer to Question #185895 in Discrete Mathematics for Aroosha ch

Question

Consider the following assertions about the sets A, B and C. Write them down in the language of predicate logic. Use only the constructions of predicate logic (∀, ∃, ¬, ⇒, ∧, ∨) and the element of symbol (∈). Do not use derived notions (∩, ∪, =, etc.).

Hint “A is a subset of B” can be formalized as ∀x. x ∈ A =⇒ x ∈ B.

(i)

(ii)

(iii) The sets A and B are equal.

Every element of A is in the set B or the set C.

If A is disjoint from B then B and C overlap.

Answer 1

Solution.

The sets A and B are equal.

$\forall x (x\isin A\implies x \in B) \land (x \in B \implies x \in A).$

Every element of A is in the set B or the set C.

$\forall x (x \in A )\implies (x\in B \lor x \in C).$

If A is disjoint from B then B and C overlap.

$\forall x ((x \in A \land x \notin B)\lor (x \in B \land x \notin A)\implies (x \in B \land x\in C).$