3. Suppose β = {1,2,3, β¦ } is the universal set and π΄π΄ = {π₯π₯: π₯π₯ β€ 6}, π΅π΅ = {π₯π₯: 4 β€ π₯π₯ β€ 9} and
πΆπΆ = {2,3,5,7,8}. Then, find π΄π΄ β© (π΅π΅ β πΆπΆ).
First, let's break down each set individually:
Set A consists of all numbers less than or equal to 6: π΄π΄ = {1, 2, 3, 4, 5, 6}.
Set B consists of all numbers between 4 and 9 (inclusive): π΅π΅ = {4, 5, 6, 7, 8, 9}.
Set C consists of the numbers 2, 3, 5, 7, and 8: πΆπΆ = {2, 3, 5, 7, 8}.
Next, let's find the symmetric difference (β) between sets B and C. The symmetric difference of two sets is the set of elements that are in either of the sets, but not in the intersection of the two sets.
The intersection of sets B and C is {5, 7, 8}. Therefore, the symmetric difference of sets B and C is {(4, 6, 9}.
Finally, let's find the intersection of set A and the symmetric difference of sets B and C.
π΄π΄ β© (π΅π΅ β πΆπΆ) = {1, 2, 3, 4, 5, 6} β© {4, 6, 9} = {4, 6}
Therefore, π΄π΄ β© (π΅π΅ β πΆπΆ) = {4, 6}.
To find π΄π΄ β© (π΅π΅ β πΆπΆ), we need to first find π΅π΅ β πΆπΆ.
1. π΅π΅ β πΆπΆ represents the symmetric difference between sets π΅π΅ and πΆπΆ. The symmetric difference of two sets is the set of elements that are in either of the sets, but not in their intersection. In other words, to find π΅π΅ β πΆπΆ, we need to remove the elements that are common to both π΅π΅ and πΆπΆ.
2. π΅π΅ = {π₯π₯: 4 β€ π₯π₯ β€ 9} represents the set of numbers between 4 and 9 (including 4 and 9).
3. πΆπΆ = {2, 3, 5, 7, 8} represents the set of numbers 2, 3, 5, 7, and 8.
4. To find the intersection of π΅π΅ and πΆπΆ, we need to identify the numbers that are common to both sets: {5, 7, 8}.
5. Now, we need to remove the common elements {5, 7, 8} from π΅π΅ to find π΅π΅ β πΆπΆ:
π΅π΅ β πΆπΆ = {4, 6}.
6. Finally, to find π΄π΄ β© (π΅π΅ β πΆπΆ), we need to find the intersection between π΄π΄ and π΅π΅ β πΆπΆ:
- π΄π΄ = {π₯π₯: π₯π₯ β€ 6} represents the set of numbers less than or equal to 6.
- π΅π΅ β πΆπΆ = {4, 6}.
The intersection between π΄π΄ and π΅π΅ β πΆπΆ is {4, 6}. Therefore, π΄π΄ β© (π΅π΅ β πΆπΆ) = {4, 6}.
To find the intersection of sets π΄π΄ and (π΅π΅ β πΆπΆ), we need to first calculate the symmetric difference (β) of sets π΅π΅ and πΆπΆ, and then find the intersection with π΄π΄.
Step 1: Find π΅π΅ β πΆπΆ
The symmetric difference of two sets, denoted by β, is the set of elements that are in either of the sets but not in their intersection.
Given π΅π΅ = {π₯π₯: 4 β€ π₯π₯ β€ 9} and πΆπΆ = {2, 3, 5, 7, 8}, we need to find the elements that are in π΅π΅ or πΆπΆ but not in both.
π΅π΅ β πΆπΆ = {π₯π₯ β π΅π΅ βͺ πΆπΆ : π₯π₯ β π΅π΅ β© πΆπΆ}
The intersection of π΅π΅ and πΆπΆ is {5, 8}, so we need to find the elements that are in π΅π΅ or πΆπΆ but not in {5, 8}.
π΅π΅ β πΆπΆ = {4, 6, 9}
Step 2: Find π΄π΄ β© (π΅π΅ β πΆπΆ)
Now that we have the symmetric difference π΅π΅ β πΆπΆ, we need to find the intersection with π΄π΄.
Given π΄π΄ = {π₯π₯: π₯π₯ β€ 6}, we are looking for the elements that are both in π΄π΄ and π΅π΅ β πΆπΆ.
π΄π΄ β© (π΅π΅ β πΆπΆ) = {π₯π₯ β π΄π΄ β© (π΅π΅ β πΆπΆ)}
Since π΄π΄ = {1, 2, 3, 4, 5, 6} and (π΅π΅ β πΆπΆ) = {4, 6, 9}, we find the common elements:
π΄π΄ β© (π΅π΅ β πΆπΆ) = {4, 6}
Therefore, the intersection of sets π΄π΄ and (π΅π΅ β πΆπΆ) is {4, 6}.