Find the number of elements in A1∪A2∪A3, supposing that there are
103 elements in A1,
992 elements in A2,
10011 elements in A3,
in each of the following situations:
(a) The sets are pairwise disjoint, that is, no pair has common elements, that is, the intersection of the two sets in any given pair is empty.
(b) A1⊆A2⊆A3.
(c) Each pairwise intersection of these sets has the same cardinality 16 , and the intersection of all three sets has cardinality 2.
a) just add them 11106
b) They are all in A3 so 10011
c) draw a Venn diagram
Only in A1 = 103 - 44 = 59
Only in A2 = 992- 44= 948
Only in A3 = 10011 - 44 = 9967
shared = 3*14 + 2 = 44
sum = 11018
should subtract 30 (14*2+2), not 44
c) draw a Venn diagram
Only in A1 = 103 - 30 = 73
Only in A2 = 992- 30= 962
Only in A3 = 10011 - 30 = 9981
shared = 3*14 + 2 = 44
sum = 11060
Thanks man!
You are welcome :)
To find the number of elements in A1∪A2∪A3 in each situation, we need to consider the given conditions for each case.
(a) In this situation, since the sets A1, A2, and A3 are pairwise disjoint, the intersection of any two sets is empty. So, the number of elements in A1∪A2∪A3 can be found by summing up the number of elements in each set individually.
Number of elements in A1∪A2∪A3 = number of elements in A1 + number of elements in A2 + number of elements in A3
= 103 + 992 + 10011
= 11006
Therefore, there are 11006 elements in A1∪A2∪A3 in situation (a).
(b) In this situation, the sets A1, A2, and A3 follow an inclusion relationship, where A1 is a subset of A2 and A2 is a subset of A3. To find the number of elements in A1∪A2∪A3, we need to consider the largest set, A3.
Number of elements in A1∪A2∪A3 = number of elements in A3
= 10011
Therefore, there are 10011 elements in A1∪A2∪A3 in situation (b).
(c) In this situation, each pairwise intersection of the sets has the same cardinality of 16, and the intersection of all three sets has cardinality 2. To find the number of elements in A1∪A2∪A3, we need to consider the following formula:
Number of elements in A1∪A2∪A3 = (sum of elements in individual sets) - (sum of elements in pairwise intersections) + (number of elements in intersection of all three sets)
Number of elements in A1∪A2∪A3 = (103 + 992 + 10011) - ((16 * 3) - 2)
= 11006 - 46
= 10960
Therefore, there are 10960 elements in A1∪A2∪A3 in situation (c).