$X, B$ and $C$ are the subset of universal set $U$. If $U =$ whole numbers less than $125$, $X =$ natural numbers less than $65, B =$ odd numbers less than 125, and $C =$ prime numbers, then:

(a) If perfect square numbers are removed from set $B$, what would be the elements of $(A \cup B) \cap C$?

Define the sets A , B , and C based on the given conditions.
Remove perfect squares from set B to obtain set B ′ .
Find the union of sets A and B ′ , denoted as A ∪ B ′ .
Find the intersection of ( A ∪ B ′ ) and C , resulting in the final set: 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 53 , 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97 , 101 , 103 , 107 , 109 , 113 ​ .

Explanation

Understanding the Problem We are given three sets: A , B , and C , which are subsets of the universal set U . We need to find the elements of ( A ∪ B ′ ) ∩ C , where B ′ is the set B with perfect squares removed. Let's break this down step by step.

Defining the Sets First, let's define the sets:


U = {whole numbers less than 125} = { 0 , 1 , 2 , ... , 124 }
A = {natural numbers less than 6} = { 1 , 2 , 3 , 4 , 5 }
B = {odd numbers less than 125} = { 1 , 3 , 5 , 7 , 9 , ... , 123 }
C = {prime numbers less than 125} = { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 53 , 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97 , 101 , 103 , 107 , 109 , 113 , 127 (127 is not less than 125, so we exclude it)}
C = { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 53 , 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97 , 101 , 103 , 107 , 109 , 113 }

Removing Perfect Squares from B Next, we need to remove the perfect square numbers from set B . The perfect squares less than 125 are 1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 , 121 . Since B only contains odd numbers, we only need to consider the odd perfect squares: 1 , 9 , 25 , 49 , 81 , 121 .

So, B ′ = B - { 1 , 9 , 25 , 49 , 81 , 121 } = { 3 , 5 , 7 , 11 , 13 , 15 , 17 , 19 , 21 , 23 , 27 , 29 , 31 , 33 , 35 , 37 , 39 , 41 , 43 , 45 , 47 , 51 , 53 , 55 , 57 , 59 , 61 , 63 , 65 , 67 , 69 , 71 , 73 , 75 , 77 , 79 , 83 , 85 , 87 , 89 , 91 , 93 , 95 , 97 , 99 , 101 , 103 , 105 , 107 , 109 , 111 , 113 , 115 , 117 , 119 , 123 }

Finding the Union of A and B' Now, we find the union of A and B ′ :

A ∪ B ′ = { 1 , 2 , 3 , 4 , 5 } ∪ { 3 , 5 , 7 , 11 , 13 , 15 , 17 , 19 , 21 , 23 , 27 , 29 , 31 , 33 , 35 , 37 , 39 , 41 , 43 , 45 , 47 , 51 , 53 , 55 , 57 , 59 , 61 , 63 , 65 , 67 , 69 , 71 , 73 , 75 , 77 , 79 , 83 , 85 , 87 , 89 , 91 , 93 , 95 , 97 , 99 , 101 , 103 , 105 , 107 , 109 , 111 , 113 , 115 , 117 , 119 , 123 }
A ∪ B ′ = { 1 , 2 , 3 , 4 , 5 , 7 , 11 , 13 , 15 , 17 , 19 , 21 , 23 , 27 , 29 , 31 , 33 , 35 , 37 , 39 , 41 , 43 , 45 , 47 , 51 , 53 , 55 , 57 , 59 , 61 , 63 , 65 , 67 , 69 , 71 , 73 , 75 , 77 , 79 , 83 , 85 , 87 , 89 , 91 , 93 , 95 , 97 , 99 , 101 , 103 , 105 , 107 , 109 , 111 , 113 , 115 , 117 , 119 , 123 }

Finding the Intersection Finally, we find the intersection of ( A ∪ B ′ ) and C :

( A ∪ B ′ ) ∩ C = { 1 , 2 , 3 , 4 , 5 , 7 , 11 , 13 , 15 , 17 , 19 , 21 , 23 , 27 , 29 , 31 , 33 , 35 , 37 , 39 , 41 , 43 , 45 , 47 , 51 , 53 , 55 , 57 , 59 , 61 , 63 , 65 , 67 , 69 , 71 , 73 , 75 , 77 , 79 , 83 , 85 , 87 , 89 , 91 , 93 , 95 , 97 , 99 , 101 , 103 , 105 , 107 , 109 , 111 , 113 , 115 , 117 , 119 , 123 } ∩ { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 53 , 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97 , 101 , 103 , 107 , 109 , 113 }
( A ∪ B ′ ) ∩ C = { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 53 , 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97 , 101 , 103 , 107 , 109 , 113 }
Examples
Understanding sets and set operations like union and intersection is crucial in many real-world scenarios. For example, in marketing, you might have set A representing customers who bought product A, set B representing customers who bought product B, and set C representing customers who responded to a marketing campaign. ( A ∪ B ) ∩ C would then represent the customers who bought either product A or B AND responded to the marketing campaign. This helps target specific customer segments for future campaigns.

Answered by GinnyAnswer | 2025-07-05

After removing perfect square numbers from set B and finding the union with set X, the intersection with the prime numbers results in the set of prime numbers less than 125: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}.
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Answered by Anonymous | 2025-07-06