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Question:

If A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; Find


(i) A ∩ B 

(ii) B ∩ C 

(iii) A ∩ C ∩ D

(iv) A ∩ C 

(v) B ∩ D 

(vi) A ∩ (B ∪ C)

(vii) A ∩ D 

(viii) A ∩ (B ∪ D) 

(ix) ( A ∩ B ) ∩ ( B ∪ C )

(x) ( A ∪ D) ∩ ( B ∪ C)

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Last updated date: 13th Jun 2024
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Answer
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Solution:

Hint: The intersection of two sets, denoted by $A \cap B$, is the set containing all elements that are common to both A and B.


The union of two sets, denoted by $A \cup B$, is the set containing all elements that belong to either A or B or both.


Step-by-Step Solution:

Given:

$A = \{ 3, 5, 7, 9, 11 \}$

$B = \{ 7, 9, 11, 13 \}$

$C = \{ 11, 13, 15 \}$

$D = \{ 15, 17 \}$


(i) A∩B

The common elements between A and B are 7, 9, 11.

$ A \cap B = \{ 7, 9, 11 \} $


(ii) B∩C

The common elements between B and C are 11, 13.

$ B \cap C = \{ 11, 13 \} $


(iii) A∩C∩D

The only common element among A, C, and D is none.

$ A \cap C \cap D = \emptyset $


(iv) A∩C

The common element between A and C is 11.

$ A \cap C = \{ 11 \} $


(v) B∩D

There is no common element between B and D.

$ B \cap D = \emptyset $


(vi) $ A \cap (B \cup C)  $

$ B \cup C = \{ 7, 9, 11, 13, 15 \} $

The common elements between A and $ B \cup C $ are 7, 9, and 11.

$ A \cap (B \cup C) = \{ 7, 9, 11 \} $


(vii) $ A \cap D  $

There is no common element between A and D.

$ A \cap D = \emptyset $


(viii) $ A \cap (B \cup D) $

$ B \cup D = \{ 7, 9, 11, 13, 15, 17 \} $

The common elements between A and $ B \cup D $ are 7, 9, and 11.

$ A \cap (B \cup D) = \{ 7, 9, 11 \} $


(ix) $ (A \cap B) \cap (B \cup C) $

Using (i) and (vi), we can say:

$ (A \cap B) \cap (B \cup C) = \{ 7, 9, 11 \} $


(x) $ (A \cup D) \cap (B \cup C) $

$A \cup D = \{ 3, 5, 7, 9, 11, 15, 17 \}$ 

$(A \cup D) \cap (B \cup C) = \{ 7, 9, 11, 15 \} $


Note: The intersection operation gives the set of elements that are common to both sets, whereas the union operation gives all unique elements from both sets. It's useful to visualize these operations with Venn diagrams, which can make it easier to understand the relationships between multiple sets.