Let A and B be the multisets {4.a, 2.b, 3.c} and {3.a, 6.b, 4.d} respectively, find (a) A U B (b) A∩B (c) A-B d) B-A

Let's solve the problem step by step, starting with the definitions of the operations on multisets.

Given:
- Multiset \( A = \{4 \cdot a, 2 \cdot b, 3 \cdot c\} \)
- Multiset \( B = \{3 \cdot a, 6 \cdot b, 4 \cdot d\} \)

### (a) \( A \cup B \) (Union of multisets):
For the union of two multisets, the multiplicity of an element in \( A \cup B \) is the maximum multiplicity of that element in either \( A \) or \( B \).

- For \( a \), multiplicity = \( \max(4, 3) = 4 \)
- For \( b \), multiplicity = \( \max(2, 6) = 6 \)
- For \( c \), multiplicity = \( \max(3, 0) = 3 \) (since \( c \) is not in \( B \))
- For \( d \), multiplicity = \( \max(0, 4) = 4 \) (since \( d \) is not in \( A \))

Thus, \( A \cup B = \{4 \cdot a, 6 \cdot b, 3 \cdot c, 4 \cdot d\} \).

### (b) \( A \cap B \) (Intersection of multisets):
For the intersection of two multisets, the multiplicity of an element in \( A \cap B \) is the minimum multiplicity of that element in either \( A \) or \( B \).

- For \( a \), multiplicity = \( \min(4, 3) = 3 \)
- For \( b \), multiplicity = \( \min(2, 6) = 2 \)
- For \( c \), multiplicity = \( \min(3, 0) = 0 \) (since \( c \) is not in \( B \))
- For \( d \), multiplicity = \( \min(0, 4) = 0 \) (since \( d \) is not in \( A \))

Thus, \( A \cap B = \{3

Learn how to find the union, intersection, and differences of multisets A and B. Detailed step-by-step solution provided for the problem involving multisets with elements {4a, 2b, 3c} and {3a, 6b, 4d}. Suitable for advanced high school students and anyone interested in set theory and multiset operations.

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