Completing a Venn Diagram with Given Set Information
This Venn diagram represents three sets: A, B, and C, and their intersections within the universal set S.
We are given the following information:
- The number of elements in set A is n(A) = 8.
- The number of elements in set B is n(B) = 16.
- The number of elements in set S is n(S) = 135.
Let's use this information to fill in the missing parts of the Venn diagram.
1. First, sum up the numbers already given inside the Venn diagram to find the number of elements in set C:
Inside C (but outside A and B) = 15
Inside C and A (but outside B) = 3
Inside C and B (but outside A) = 10
Inside C, A, and B = 4
Total so far = 15 + 3 + 10 + 4 = 32
2. Since we don't have n(C) given explicitly, we cannot directly fill the value for C yet. Instead, let's consider set A and set B and their overlap.
For A, apart from the 3 (exclusive to A and C) and the 4 (in the intersection of A, B, and C), there must be another 1 element which is exclusive to A (since 8 elements are in A in total): 3 + 4 + 1 = 8.
Now, let's update the diagram's inside A but outside B and C to be 1.
3. Moving on to B, there is already 4 inside the intersection with A and C, and 10 in the intersection with C, meaning we need to find how many elements are exclusively in B by subtracting the known elements inside B from the total number in B, n(B)= 16.
Total number in B = 16, and we know the intersections:
Inside B and A = 4
Inside B and C = 10
Exclusive to B = 16 - (Intersection B and A + Intersection B and C) = 16 - (4 + 10) = 2.
Now, let's update the diagram's inside B but outside A and C to be 2.
4. Now, we can find out the number of elements exclusive to C by subtracting the known elements in intersections involving C from what we've calculated for set C:
Exclusive to C = (Total so far in C) - (Inside C and A) - (Inside C and B) - (Inside C, A, and B)
Exclusive to C = 32 - 3 - 10 - 4 = 15.
But this value was already filled in the diagram within C but outside A and B, so our previous calculations are consistent.
5. Lastly, fill in the value of the universal set S, which includes all the elements in sets A, B, C, and also those that are not in any of these sets:
Total elements in the sets A, B, and C (already filled in diagram including all intersections) = 1 + 3 + 4 + 10 + 2 + 15 = 35.
Hence, the number of elements not in sets A, B, or C but within the universal set S = n(S) - total in A, B, C = 135 - 35 = 100.
This last value of 100 is also already present in the diagram, which indicates the number of elements outside sets A, B, and C but within S.
With all numbers filled in the Venn diagram, each subset within the universal set is accounted for.
