Question

The Set A has 4 elements and the Set B has 5 elements then the number of injective mappings that can be defined from A to B is:
A. $144$
B. $72$
C. $60$
D. $120$

Answer 1

A General Function points from each member of "A" to a member of "B". It never has one "A" pointing to more than one "B", so one-to-many is not OK in a function (so something like "f(x) = 7 or 9" is not allowed)But more than one "A" can point to the same "B" (many-to-one is OK).Injective means we won't have two or more "A"s pointing to the same "B". So many-to-one is NOT OK (which is OK for a general function). As it is also a function one-to-many is not OK But we can have a "B" without a matching "A" Injective is also called "One-to-One".

Complete step by step solution: The Set A has 4 elements and the Set B has 5 elements and we have to find the number of injective mappings.
Let f be such a function.
Now, f takes inputs from set A whereas the output value of f comes from set B.
Using the fact that injective functions are one-one and onto,
$f(1)\;$can take $5$ values,
$\;f(2)\;$can then take only $4\;$values ,
$\;f(3)\;$can take only $3\;$and
$\;f(4)\;$only $2$.
Hence the total number of functions are $5 \times 4 \times 3 \times 2 = 120$.

Note: In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements of its co-domain. In simple words, every element of the function’s co-domain is the image of at most one element of its domain.