a)Consider the formula n (A U B) = n (A) + - n (A �¿ B).
(a) Show that this relation holds for A = {1, 2, 3, 4} and B= {2, 4, 5, 6, 7, 8}
(b) Make up your own two sets A and B, each consisting of at least six elements. Using these two sets, show that the relationship above holds
(a) To show that the formula n(A U B) = n(A) + n(A �¿ B) holds for A = {1, 2, 3, 4} and B = {2, 4, 5, 6, 7, 8}, we need to calculate the cardinalities of the sets involved and see if they satisfy the equation.
n(A) = the number of elements in set A
= 4 (since A has 4 elements)
n(B) = the number of elements in set B
= 6 (since B has 6 elements)
n(A U B) = the number of elements in the union of sets A and B
= the number of elements in {1, 2, 3, 4, 5, 6, 7, 8}
= 8 (since there are 8 distinct elements)
n(A �¿ B) = the number of elements in the set A without the common elements with B
= the number of elements in {1, 3}
= 2 (since there are 2 elements)
Now, let's plug in these values into the formula and check if it holds:
n(A U B) = n(A) + n(A �¿ B)
= 4 + 2
= 6
Since n(A U B) = 8 and n(A) + n(A �¿ B) = 6, the relation n(A U B) = n(A) + n(A �¿ B) holds for A = {1, 2, 3, 4} and B = {2, 4, 5, 6, 7, 8}.
(b) To show that the formula n(A U B) = n(A) + n(A �¿ B) holds for your own two sets A and B, each consisting of at least six elements, you need to follow a similar process.
1. Choose two sets A and B, each with at least six elements.
2. Calculate the cardinalities of A and B, denoted as n(A) and n(B), respectively.
3. Calculate the cardinality of the union of A and B, denoted as n(A U B).
4. Calculate the cardinality of the set A without the common elements with B, denoted as n(A �¿ B).
5. Plug in the values of n(A), n(B), n(A U B), and n(A �¿ B) into the formula n(A U B) = n(A) + n(A �¿ B).
6. Check if the equation holds by comparing the left and right sides of the equation.
Make sure to choose sets A and B and perform the calculations according to the specific values you have.