If X={1,2,3,4,5} and Y={a,b,c,d,e,f} and `f:X rarr Y`, find the total number of
`{:((i)" functions ",(ii)" one to one functions "),((iii)" many-one functions ",(iv)" constant functions "),((v)" onto functions ",(vi)" into functions "):}`

Answer

To solve the problem step by step, we will analyze the given sets and apply the relevant formulas for each type of function.

### Given:
- Set \( X = \{1, 2, 3, 4, 5\} \) (Domain, \( m = 5 \))
- Set \( Y = \{a, b, c, d, e, f\} \) (Co-domain, \( n = 6 \))

### (i) Total Number of Functions
The total number of functions from set \( X \) to set \( Y \) is given by the formula:
\[ \text{Total Functions} = n^m \]
Substituting the values:
\[ \text{Total Functions} = 6^5 = 7776 \]

### (ii) One-to-One Functions
For one-to-one functions, we can use the formula:
\[ \text{One-to-One Functions} = P(n, m) = \frac{n!}{(n-m)!} \]
Where \( P(n, m) \) is the number of permutations of \( n \) items taken \( m \) at a time. Here \( n = 6 \) and \( m = 5 \):
\[ \text{One-to-One Functions} = \frac{6!}{(6-5)!} = \frac{6!}{1!} = 720 \]

### (iii) Many-One Functions
Many-one functions can be calculated by subtracting the number of one-to-one functions from the total number of functions:
\[ \text{Many-One Functions} = \text{Total Functions} - \text{One-to-One Functions} \]
Substituting the values:
\[ \text{Many-One Functions} = 7776 - 720 = 7056 \]

### (iv) Constant Functions
The number of constant functions is equal to the number of elements in the co-domain \( Y \):
\[ \text{Constant Functions} = n = 6 \]

### (v) Onto Functions
An onto function requires that every element of the co-domain \( Y \) is mapped by at least one element of the domain \( X \). Since \( m < n \) (5 < 6), there cannot be any onto functions:
\[ \text{Onto Functions} = 0 \]

### (vi) Into Functions
Into functions are defined as functions that do not cover the entire co-domain. The number of into functions can be calculated as:
\[ \text{Into Functions} = \text{Total Functions} - \text{Onto Functions} \]
Substituting the values:
\[ \text{Into Functions} = 7776 - 0 = 7776 \]

### Summary of Results:
1. Total Functions: 7776
2. One-to-One Functions: 720
3. Many-One Functions: 7056
4. Constant Functions: 6
5. Onto Functions: 0
6. Into Functions: 7776