Refer to the following Venn diagram. 12 A B 15 20 U Find the following. (a) n(A n B) (b) n(A° n Bº) (c) n[(A n B)°] (d) n(A° U Bº) (e) n[(A n Bº) U (A°n B)] (f) n(U^)
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
only need parts (d) (e) and (f)
![### Educational Resource on Venn Diagrams
#### Venn Diagram Representation
The Venn diagram contains two intersecting circles labeled \(A\) and \(B\), set inside a universal set \(U\). The elements are distributed as follows:
- Set \(A\) contains two regions: one with 12 elements and the intersection with \(B\) that has 3 elements.
- Set \(B\) contains two regions: one with 15 elements and the intersection with \(A\) that has 3 elements.
- The region outside both sets \(A\) and \(B\), within the universal set \(U\), contains 20 elements.
#### Exercises
Find the following using the Venn diagram:
(a) \( n(A \cap B) \)
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(b) \( n(A^c \cap B^c) \)
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(c) \( n[(A \cap B)^c] \)
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(d) \( n(A^c \cup B^c) \)
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(e) \( n[(A^c \cap B^c) \cup (A^c \cap B)] \)
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(f) \( n(U^c) \)
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#### Explanation of Terms
- \( n(A \cap B) \): Number of elements in the intersection of sets \(A\) and \(B\).
- \( A^c \): Complement of set \(A\), elements not in \(A\).
- \( B^c \): Complement of set \(B\), elements not in \(B\).
- \( n(A^c \cap B^c) \): Number of elements in the intersection of complements of \(A\) and \(B\).
- \( n[(A \cap B)^c] \): Number of elements not in the intersection of \(A\) and \(B\).
- \( n(A^c \cup B^c) \): Number of elements in the union of complements of \(A\) and \(B\).
- \( n(U^c) \): Number of elements outside the universal set \(U\), typically zero if \(U\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6e5354a-af81-4083-9a8b-a2eadb950c57%2F48ca1e6d-4f82-4f69-b98c-507350c718f5%2Frk926g_processed.png&w=3840&q=75)
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