Find the number m of elements in the union of sets A, B, C, D where:(i) A, B, C, D have 50, 60, 70, 80 elements, respectively.(ii) Each pair of sets has 20 elements in common.(iii) Each three of the sets has 10 elements in common.(iv)All four of the sets have 5 elements in common.
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To find the number of elements in the union of sets A, B, C, D, we need to consider the total number of elements in each set and any common elements that exist between the sets. Here's how we can calculate it:
(i) The individual size of each set:
Set A has 50 elements.
Set B has 60 elements.
Set C has 70 elements.
Set D has 80 elements.
(ii) Each pair of sets has 20 elements in common:
Since each pair of sets has 20 elements in common, we can calculate the total number of common elements as follows:
Number of elements in common between sets A and B = 20
Number of elements in common between sets A and C = 20
Number of elements in common between sets A and D = 20
Number of elements in common between sets B and C = 20
Number of elements in common between sets B and D = 20
Number of elements in common between sets C and D = 20
(iii) Each three of the sets has 10 elements in common:
Since each three sets have 10 elements in common, we can calculate the total number of common elements as follows:
Number of elements in common between sets A, B, and C = 10
Number of elements in common between sets A, B, and D = 10
Number of elements in common between sets A, C, and D = 10
Number of elements in common between sets B, C, and D = 10
(iv) All four of the sets have 5 elements in common:
Since all four sets have 5 elements in common, we can calculate the total number of common elements as follows:
Number of elements in common between sets A, B, C, and D = 5
Now let's calculate the number of elements in the union of sets A, B, C, D:
Step 1: Start with the sum of the sizes of each individual set:
Total elements in the union = Size of set A + Size of set B + Size of set C + Size of set D
Total elements in the union = 50 + 60 + 70 + 80
Total elements in the union = 260
Step 2: Subtract the common elements counted more than once:
We need to subtract the total count of common elements between each pair of sets, each three sets, and all four sets:
Total elements in the union = 260 - (Common elements between pairs) - (Common elements between three sets) - (Common elements between all four sets)
Total elements in the union = 260 - (6 * 20) - (4 * 10) - 5
Total elements in the union = 260 - 120 - 40 - 5
Total elements in the union = 95
Therefore, the number of elements in the union of sets A, B, C, D is 95.