$\pi, B$ and $C$ are the subset of universal set
$U$. If $U=S$ whole number less than $22 S$, $\pi=\Sigma$ natural number less than $6 S, B=$ God number less than $I Z S$ and $C=$ Sprime numbers.
(a) If perfect square number is removed from set $B$, what would the elements of (ZUB) $\cap C$ ?
Answer (1)
Define the sets U , π , B , and C based on the given conditions and assuming S = 1 and I Z = 1 .
Remove perfect square numbers from set B , resulting in B ′ = { } .
Find the union of sets π and B ′ , which gives π ∪ B ′ = { 1 , 2 , 3 , 4 , 5 } .
Find the intersection of the set ( π ∪ B ′ ) and C , resulting in ( π ∪ B ′ ) ∩ C = { 2 , 3 , 5 } .
The elements of ( π ∪ B ) ∩ C are { 2 , 3 , 5 } .
Explanation
Problem Analysis and Set Definitions We are given several sets: U , π , B , and C . We are asked to find the elements of ( π ∪ B ) ∩ C after removing perfect squares from set B . Let's define each set based on the problem statement, assuming that 'Z' is a typo and should be ' π '. Also, we need to assume values for S and I Z to proceed with a concrete example. Let's assume S = 1 and I Z = 1 .
Defining the Sets with S=1 and IZ=1 Given S = 1 and I Z = 1 :
U = Whole numbers less than 22 S = 22 ( 1 ) = 22 . So, U = { 0 , 1 , 2 , 3 , ... , 21 } .
π = Natural numbers less than 6 S = 6 ( 1 ) = 6 . So, π = { 1 , 2 , 3 , 4 , 5 } .
B = God numbers less than I ZS = 1 ( 1 ) = 1 . So, B = { } . Since the problem states 'God numbers', and the range is less than 1, we interpret this as an empty set because there are no positive integers less than 1.
C = Prime numbers less than 22 S = 22 ( 1 ) = 22 . So, C = { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 } .
Removing Perfect Squares from B Now, we remove perfect squares from set B . Since B is an empty set, removing perfect squares still leaves it as an empty set. So, B ′ = { } .
Finding the Union of Pi and B' Next, we find the union of π and B ′ : π ∪ B ′ = { 1 , 2 , 3 , 4 , 5 } ∪ { } = { 1 , 2 , 3 , 4 , 5 } .
Finding the Intersection with C Finally, we find the intersection of ( π ∪ B ′ ) and C : ( π ∪ B ′ ) ∩ C = { 1 , 2 , 3 , 4 , 5 } ∩ { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 } = { 2 , 3 , 5 } .
Final Answer Therefore, the elements of ( π ∪ B ) ∩ C after removing perfect squares from set B are { 2 , 3 , 5 } .
Examples
Understanding sets and their operations like union and intersection is crucial in many real-world scenarios. For instance, consider a school club where π represents students interested in mathematics, B represents students interested in biology, and C represents students who are athletes. Finding ( π ∪ B ) ∩ C would identify the athletes who are interested in either mathematics or biology, helping the school organize specific programs tailored to this group. This kind of set analysis is also used in database management, data analysis, and even in designing search algorithms.
