Please help me with this.Use element argument to prove the statement. Assume that all sets are subsets of a universal set U, Statement: For all sets A, B, and C, A ∩ (B − C) = (A ∩ B) − (A ∩ C). Please do the proof like the way the example proof does it. Thanks. Statement: For all sets A, B, and C,(A - 8)n (C - 8) - (ANC) - 8.Proof:Suppose A, B, and C are any sets. [To show that (A - 8) n (C - 8) - (ANC) - 8, we must show that (A -8)n (C - 8) C (ANC) - B and that (ANC) - 8S ( - 8) n (C - 8).1Part 1: Proof that (A - B) n (C - 8) S (ANC) - 8Consider the sentences in the following scrambled list.By definition of set difference, xCA and x EB and xeC and xe 8.Thus x EAN C by definition of intersection, and, in addition, xe 8.Therefore x c (ANg-8 by the definition of set difference.By definition of intersection, xEA and x B and x ECand xe8.By definition of set difference, xeA-B and xEC-B.By definition of intersection, xEA -B and xEC-8.To prove Part 1, select sentences from the list and put them in the correct order.1. Suppose x€ (A - B) n (C- 8).2. By defintion of intersection, xe A-Band xeC-B3. By defintion of set diferencereA and xe B and xeCand KEOV4. Thus xeANC by definition of intersection and in addition e6 v5. ThereforexE (ANC-B by the defintion of set diference6. Hence, (A -8) n (c - 8) C(ANg-8 by definition of subset.Part 2: Proof that (ANC) - BE (A - 8) n (C - B)Consider the sentences in the following scrambled list.By definition of set deference x EAN C and x B.By definition of intersection XEANCand xEB.By definition of set dfference, xEA and xEC.Thus, by definition of intersection, EA and xEC, and, in addition, xe 8.So by definition of set difference, xEA-B and xEC-B.Hence both x EA and xeB and also xe C, and xB.By definition of intersection, xE(A-8)n (C- 8).To prove Part 2, select sentences from the list and put them in the correct order.1. Suppose xE (ANC)-8.2. by detinton of set diterencexCANC and xeB3. Thus, by definition of intersection xe A and xeC and, in addition, eb.4. (Hence both xCA and xeB andEsoxeCandeE5. So by defintion of set diferenceXEA-Band xeC-86. [By defnton of intersection xE A-BNIC-B)7. Hence, (ANC) -BE (A - 0) n (C - 8) by definition of subset.Conclusion:

Question

Please help me with this.Use element argument to prove the statement. Assume that all sets are subsets of a universal set U, Statement: For all sets A, B, and C, A ∩ (B − C) = (A ∩ B) − (A ∩ C). Please do the proof like the way the example proof does it. Thanks. Statement: For all sets A, B, and C,(A - 8)n (C - 8) - (ANC) - 8.Proof:Suppose A, B, and C are any sets. [To show that (A - 8) n (C - 8) - (ANC) - 8, we must show that (A -8)n (C - 8) C (ANC) - B and that (ANC) - 8S ( - 8) n (C - 8).1Part 1: Proof that (A - B) n (C - 8) S (ANC) - 8Consider the sentences in the following scrambled list.By definition of set difference, xCA and x EB and xeC and xe 8.Thus x EAN C by definition of intersection, and, in addition, xe 8.Therefore x c (ANg-8 by the definition of set difference.By definition of intersection, xEA and x B and x ECand xe8.By definition of set difference, xeA-B and xEC-B.By definition of intersection, xEA -B and xEC-8.To prove Part 1, select sentences from the list and put them in the correct order.1. Suppose x€ (A - B) n (C- 8).2. By defintion of intersection, xe A-Band xeC-B3. By defintion of set diferencereA and xe B and xeCand KEOV4. Thus xeANC by definition of intersection and in addition e6 v5. ThereforexE (ANC-B by the defintion of set diference6. Hence, (A -8) n (c - 8) C(ANg-8 by definition of subset.Part 2: Proof that (ANC) - BE (A - 8) n (C - B)Consider the sentences in the following scrambled list.By definition of set deference x EAN C and x B.By definition of intersection XEANCand xEB.By definition of set dfference, xEA and xEC.Thus, by definition of intersection, EA and xEC, and, in addition, xe 8.So by definition of set difference, xEA-B and xEC-B.Hence both x EA and xeB and also xe C, and xB.By definition of intersection, xE(A-8)n (C- 8).To prove Part 2, select sentences from the list and put them in the correct order.1. Suppose xE (ANC)-8.2. by detinton of set diterencexCANC and xeB3. Thus, by definition of intersection xe A and xeC and, in addition, eb.4. (Hence both xCA and xeB andEsoxeCandeE5. So by defintion of set diferenceXEA-Band xeC-86. [By defnton of intersection xE A-BNIC-B)7. Hence, (ANC) -BE (A - 0) n (C - 8) by definition of subset.Conclusion:>>>>>

Accepted Answer

Answered: Image /qna-images/answer/a7edf0f2-fa41-430c-83e9-3fbdf31d810a.jpg

Please help me with this. Use element argument to prove the statement. Assume that all sets are subsets of a universal set U, Statement: For all sets A, B, and C, A ∩ (B − C) = (A ∩ B) − (A ∩ C). Please do the proof like the way the example proof does it. Thanks.