Question

Prove or disprove that:
i) A-B⊆A-C-B
ii)(A-B)∪(B-C)∪(C-A)=(A∪B∪C)-(A∩B∩C)

Answer 1

Prove or disprove that:

1. $A - B\subseteq A - C - B$

2. $(A - B)\cup (B - C)\cup (C - A) = (A\cup B\cup C) - (A\cap B\cap C)$

Solution:

1. Answer: That is wrong.

Let $A$ be $\{1,2,3\}$ , $B = \{1\}$ , $C = \{2\}$ . $A - B = \{2,3\}$ , $A - C - B = \{3\}$ . $\{2,3\} \not\subseteq \{3\}$ .

2. Answer: That is true.

Let's denote $A_{1} = A - B - C$ , $B_{1} = B - A - C$ , $C_{1} = C - A - B$ , $AB = A \cap B - C$ , $AC = A \cap C - B$ , $BC = B \cap C - A$ , $ABC = A \cap B \cap C$ .

$A - B = A_{1} \cup AC, B - C = B_{1} \cup AB, A - C = A_{1} \cup BC \Rightarrow (A - B) \cup (B - C) \cup (C - A) = A_{1} \cup AC \cup B_{1} \cup AB \cup A_{1} \cup BC.$

$(A \cup B \cup C) - (A \cap B \cap C) = A_{1} \cup B_{1} \cup C_{1} \cup AB \cup BC \cup AC.$

Thus $(A - B)\cup (B - C)\cup (C - A) = (A\cup B\cup C) - (A\cap B\cap C)$

Answer provided by https://www.AssignmentExpert.com