Let U - 1- 2- 3- 4- 5- 6- 7- 8- 9- A - 1- 2- 3- 4- B - 2- 4- 6- 8 and C - 3- 4- 5- 6-Find i A- ii B- iii A - C-iv A - B- v A- vi B- C-

(i) Given : U = nbsp;{1, 2, 3, 4, 5, 6, 7, 8, 9}, A = nbsp;{1, 2, 3, 4} ∵ A' = U A= nbsp;{1, 2, 3, 4, 5, 6, 7, 8, 9 } nbsp;{1, 2, 3, 4} ∴ A' = nbsp ...

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Byju's Answer

Standard X

Mathematics

Intersection of Sets

Let U = 1, 2,...

Question

Let $U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8}$ and $C = {3, 4, 5, 6} .$
Find $(i) A^{'} (i i) B^{'} (i i i) (A \cup C)^{'}$
$(i v) (A \cup B)^{'} (v) (A^{'})^{'} (v i) (B - C)^{'}$

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Solution

$(i)$ Given : $U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}$
$∵ A^{'} = U - A = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4}$
$∴ A^{'} = {5, 6, 7, 8}$

$(i i)$ Given : $U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, B = {2, 4, 6, 8}$
$∵ B^{'} = U - B = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8}$
$∴ B^{'} = {1, 3, 5, 7, 9}$

$(i i i)$ Given : $U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}$ and $C = {3, 4, 5, 6}$
Step $1 :$ Finding $A \cup C$
$∴ A \cup C = {1, 2, 3, 4, 5, 6}$
Step $2 :$ Finding $(A \cup C)^{'}$
$(A \cup C)^{'} = U - (A \cup C)^{'} = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 5, 6}$
$= {7, 8, 9}$
Final answer:
$(A \cup C)^{'} = {7, 8, 9}$

$(i v)$ Given : $U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}$ and $B = {2, 4, 6, 8}$
Step $1 :$ Finding $A \cup B$
$∴ A \cup B = {1, 2, 3, 4, 6, 8}$
Step $2 :$ Finding $(A \cup B)^{'}$
$(A \cup B)^{'} = U - (A \cup B) = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 6, 8}$
$= {5, 7, 9}$
Final answer:
$(A \cup B)^{'} = {5, 7, 9}$

$(v)$ Given: $U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4,}$
Step 1: Finding complement of set $A$
$A^{'} = U - A = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4}$
$= {5, 6, 7, 8, 9}$
Step $2 :$ Finding coplement of $A^{'}$
$(A^{'})^{'} = U - A^{'} = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {5, 6, 7, 8, 9}$
$= {1, 2, 3, 4}$
$(A^{'})^{'} = A$
Final answer :
$(A^{'})^{'} = {1, 2, 3, 4} = A$

$(v i)$ Given: $U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, B = {2, 4, 6, 8}$ and $C = {3, 4, 5, 6}$
Step $1 :$ Finding $B - C$ .
$B - C = B - (B \cap C)$
$∵ B \cap C = {2, 4, 6, 8} \cap {3, 4, 5, 6} = {4, 6}$
$∴ B - C = {2, 4, 6, 8} - {4, 6}$
$= {2, 8}$
Step $2 :$ Finding complement of $(B - C)$
$(B - C)^{'} = U - (B - C)$
$= {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 8}$
$= {1, 3, 4, 5, 6, 7, 9}$

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