Given-that-A-0-1-3-5-B-1-2-4-7-and-C-1-2-3-5-8-prove-that-i-A-B-C-A-B-C-ii-A-B-C-A-B-C-iii-A-B-C-A-C-B-C-iv-A-C-B-A-B-C-B-

Question Number 93567 by oustmuchiya@gmail.com last updated on 13/May/20

G i v e n t h a t A = {0, 1, 3, 5} B = {1, 2, 4, 7} a n d C = {1, 2, 3, 5, 8} p r o v e t h a t

(i) (A \cap B) \cap C = A \cap (B \cap C)

(ii) (A \cup B) \cup C = A \cup (B \cup C)

(iii) (A \cup B) \cap C = (A \cup C) \cup (B \cap C)

(iv) (A \cap C) \cup B = (A \cup B) \cap (C \cup B)

Commented by Ritu Jain last updated on 13/May/20

(i) (A \cap B) \cap C = A \cap (B \cap C)

(A \cap B) = {0, 1, 3, 5} \cap {1, 2, 3, 7} = {1}

LHS = (A \cap B) \cap C = ({1}) \cap {1, 2, 3, 5, 8} = {1}

B \cap C = {1, 2, 4, 7} \cap {1, 2, 3, 5, 8} = {1}

RHS = A \cap (B \cap C) = {0, 1, 3, 5} \cap ({1} = {1}

LHS = RHS

Answered by otchereabdullai@gmail.com last updated on 14/May/20

ii . (AUB) = {0, 1, 2, 3, 4, 5, 7}

C = {1, 2, 3, 5, 8}

∴ {AUB} UC = {0, 1, 2, 3, 4, 5, 7, 8}

Also

A = {0, 1, 3, 5}

{BUC} = {1, 2, 3, 4, 5, 7, 8}

∴ AU (BUC) = {0, 1, 2, 3, 4, 5, 7, 8}

Now since

(AUB) UC = AU (BUC) = {0, 1, 2, 3, 4, 5, 7, 8}

we can conclude that

(AUB) UC = AU (BUC) .

iii . (AUB) \cap C = (AUC) U (B \cap C)

(AUB) = {0, 1, 2, 3, 4, 5, 7}

C = {1, 2, 3, 5, 8}

∴ (AUB) \cap C = {0, 1, 2, 3, 4, 5, 7} \cap {1, 2, 3, 5, 8}

= {1, 2, 3, 5}

also

(AUC) = {0, 1, 2, 3, 5, 8}

(B \cap C) = {1, 2}

∴ (AUC) U (B \cap C) = {0, 1, 2, 3, 5, 8}

\Rightarrow (AUB) \cap C \neq (AUC) U (B \cap C)