Question

d) We prove the stateneat ⟨D⟩ : (D) Let A,B,C be sets. Sippose A⊂C and B⊂C. Then A⊂B itf C\B⊂C\A. Let A,B,C be sets. Suppose A⊂C and B⊂C. We wast to deduce if A⊂B then C,B⊂C}A. . Suppree A⊂B. Under thits sssumption, we verify C\B⊂C\A. It is: "for aup object x, if x∈C\B then x∈C\(A.' Pick any objeet x. (I) (IV) Then, A⊂B, we would bave. (VT) by the definition of sabet ralation. Therefore Hence, to the definition of eomplecient, we have It follows that C\B∈C, B. We waat to deduce "if C\B⊂C\A then A⊂B ?] Suppoen C\BCC\A. (X) We deduoe that x∈B with the method of proof-in-eoutraliction: Then, ninee x∈C\B and C\B⊂C\A, we have le the definition of wubet relation: Then, by the definition of (XVII) , we have (XVIII) In particular, XIX ftecall that by assemption, x∈A. Then Contradiction asivek. Hence, in the fint place, we huve x∈B. It follows that A∈B, Heace A⊂B if C\B⊂C\A. Mank) (a) We jeun the alatrinevt (a) (i) Let A al be mati. Supgre iu at C at. Tarw 4 at −1. (4) Wr peur the ollatriest (aif); (4) We peime the alatianed (c).

Answer 1

To prove the statement (D) Let A,B,C be sets. Suppose A⊂C and B⊂C. Then A⊂B if C\B⊂C\A, we need to show that if A⊂B, then C\B⊂C\A.

Step-by-step explanation:

In the given question, we are asked to prove the statement ⟩D⟪: (D) Let A,B,C be sets. Suppose A⊂C and B⊂C. Then A⊂B if C\B⊂C\A. To prove this, we need to show that if A⊂B, then C\B⊂C\A.

Let's start by assuming A⊂B. Now, let's consider an arbitrary object x. If x∈C\B, then x∈C\(A∪B). This is because A⊂B implies A∪B = B, and applying the definition of set difference. Therefore, we have shown that if A⊂B, then C\B⊂C\A.