88. The total number of mappings from the set {1, 2} to the set {3, 4, 5, 6, 7} is _____.
The Total Number of Mappings from the Set {1, 2} to the Set {3, 4, 5, 6, 7}
Correct Answer
Answer: 25
Detailed Solution
Let the domain be
A = {1, 2}
and the codomain be
B = {3, 4, 5, 6, 7}.
The number of elements in the domain is
|A| = 2
and the number of elements in the codomain is
|B| = 5.
In a mapping (function), every element of the domain must be assigned to exactly one element of the codomain. The choices made for one element of the domain are completely independent of the choices made for the other elements.
The first element of the domain, namely 1, can be mapped to any one of the five elements of the codomain.
Similarly, the second element, namely 2, can also be mapped independently to any one of the same five elements.
Therefore, the total number of possible mappings is obtained by multiplying the number of choices for each domain element.
Total mappings = 5 × 5
= 5²
= 25
Hence, there are 25 different mappings from the set {1, 2} to the set {3, 4, 5, 6, 7}.
General Formula for Total Mappings
If a set A contains m elements and a set B contains n elements, then the total number of mappings (functions) from A to B is
nm
This formula works because each of the m elements of the domain has n independent choices in the codomain.
In this problem,
m = 2
n = 5
Therefore,
Total mappings = 5² = 25.
Concept Behind the Question
A mapping, also known as a function, assigns every element of the domain to exactly one element of the codomain. Different domain elements may map to the same codomain element, making many-to-one mappings possible. Since each domain element can independently choose any codomain element, the multiplication principle is applied, resulting in the formula nm. This concept forms the foundation of discrete mathematics, combinatorics, and set theory and is repeatedly tested in competitive examinations.
Final Answer
Total number of mappings = 25
Answer: 25
