Question

Roblem 4 from Section 4.1: "Let, A, B, C be sets. Write the following in "if... then" format and prove:

(a) x ∈ B is a sufficient condition for x ∈ A ∪ B.
(b) A ∩ B ∩ C = ∅ is a necessary condition for A ∩ B = ∅.
(c) A ∪ B = B is a necessary and sufficient condition for A ⊆ B."
(2) Problem 4 from Section 4.2: "Use previously proven theorems to prove the following. (a) A∩(B∩C)c =(A∩Bc)∪(A∩Cc)
(b) A∩(B∩(A∩B)c)=∅
(c) (A∩B)∪Bc =A∪Bc
(d) A∪(B−C)=(A∪B)−(C−A)."

Answer 1

In mathematics, 'if... then' statements are used to establish logical connections between conditions and consequences, particularly in the context of set theory. The examples provided illustrate the application of these statements to prove specific conditions related to set operations.

Step-by-step explanation:

When dealing with sets and conditions, it's essential to understand if... then statements and how they pertain to logical reasoning in mathematics. In the context of the provided problems:

• (a) If x ∈ B, then x ∈ A ∪ B can be proven because the definition of union includes all elements that are in either set A or B, or both.
• (b) If A ∩ B ∩ C = ∅, then A ∩ B = ∅ is a necessary condition because if there are no elements common to all three sets, there cannot be elements common to just A and B.
• (c) A ∪ B = B is a necessary and sufficient condition for A ⊆ B because if the union of A and B is B, all elements of A must be contained within B, satisfying both the necessary and sufficient condition.