Discrete Mathematical Structures by Pearson

3-4 Discrete Mathematical Structures

3.5.1 Identity Function

Identity function is a function on a set to itself that is denoted by I and is defined by,

I: A → A, where I (x) = x for every x ∈ A

that is, every element is mapped to itself. The identity function on a set A is also denoted by I

A

.

Example: Let A = {a, b, c, d}, then I

A

: A → A is defined by I(a) = a, I(b) = b, I(c) = c, I(d) = d.

a

b

c

d

a

f

b

c

d

B

A

3.5.2 One-One and Many-One Function

A function f: A → B is said to be one-one (or injective), if different elements in A have different

images in B. That is, if f (x

1

) = f (x

2

) ⇒ x

1

= x

2

, x

1

, x

2

∈ A or f (x

1

) ≠ f (x

2

) ⇒ x

1

≠ x

2

, x

1

, x

2

∈ A.

If two or ...