If the universal set U has 100 elements, set A has 40 elements, and set B has 30 elements with 10 elements in both A and B, how many elements are in neither A nor B?
Practice Questions
1 question
Q1
If the universal set U has 100 elements, set A has 40 elements, and set B has 30 elements with 10 elements in both A and B, how many elements are in neither A nor B?
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Using the principle of inclusion-exclusion, the number of elements in either A or B is: (A + B - Both) = 40 + 30 - 10 = 60. Therefore, elements in neither = U - (A ∪ B) = 100 - 60 = 40.
Questions & Step-by-step Solutions
1 item
Q
Q: If the universal set U has 100 elements, set A has 40 elements, and set B has 30 elements with 10 elements in both A and B, how many elements are in neither A nor B?
Solution: Using the principle of inclusion-exclusion, the number of elements in either A or B is: (A + B - Both) = 40 + 30 - 10 = 60. Therefore, elements in neither = U - (A ∪ B) = 100 - 60 = 40.
Steps: 7
Step 1: Identify the total number of elements in the universal set U, which is 100.
Step 2: Identify the number of elements in set A, which is 40.
Step 3: Identify the number of elements in set B, which is 30.
Step 4: Identify the number of elements that are in both sets A and B, which is 10.
Step 5: Use the principle of inclusion-exclusion to find the number of elements in either set A or set B. This is calculated as: (Number of elements in A) + (Number of elements in B) - (Number of elements in both A and B). So, it is 40 + 30 - 10 = 60.
Step 6: Now, to find the number of elements that are in neither set A nor set B, subtract the number of elements in either A or B from the total number of elements in the universal set U. This is calculated as: (Total elements in U) - (Elements in either A or B). So, it is 100 - 60 = 40.
Step 7: Therefore, the number of elements that are in neither set A nor set B is 40.
