One-to-One (Injective) Function
An injective function is a function that maps distinct elements of its domain to distinct elements of its co-domain. A function ƒ : A → B is said to be one-to-one function if every element of A has distinct image in B. We can also say that ƒ is one-to-one if ƒ(a) = ƒ(a') implies a = a'.
If A and B are two finite sets, then one-to-one from A → B is possible only if |A| ≤ |B|.
If |A| = n and |B| = m, then total number of one-to-one functions possible from A → B = mPn = P(m, n)
and total number of one-to-one functions possible from B → A = nPm = P(n, m). If |A| = |B| = n, then total number of functions possible = n!.
Onto (Surjective) Function
A function in which every element of co-domain set has at least one pre-image. A function ƒ : A → B is said to be onto if and only if every element of B is mapped by at least one element of A.
In this, range of function is equal to co-domain of function. If A and B are two finite sets, then onto function from A → B is possible only if |B| ≤ |A|.
If |A| = n and |B| = m, then total number of onto functions possible from A → B = mn - mC1(m-1)n + mC2(m-2)n - mC3(m-3)n + ..... + (-1)m mCn-11n
Bijective or Invertible Function
A bijective function is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements in each set and have only one image or only one pre-image.
A function ƒ : A → B is said to be bijective if and only if ƒ is both one-to-one(injective) and onto(surjective). Bijection from A to B is possible only if |A| = |B|. If |A| = |B| = n, then total number of bijective function = n!.
Some more Functions
Into Function
A function in which there must be an element of co-domain which does not have a pre-image in domain.
One-to-One Into Function
A function in which all the elements of domain have a different unique images in co-domain, and also there must be an element of co-domain which does not have a pre-image in domain. A function is one-to-one into if and only if it is both one-to-one(injective) and into functions.
Many-One Function
A function in which there exists two or more than two different elements in domain having the same image in co-domain.
Many-One Into Functions
A function in which there exists two or more than two different elements in domain having the same image in co-domain, and also there must be an element of co-domain which does not have a pre-image in domain. A function is many-one into if and only if it is both many-one and into functions.
Many-One Onto Functions
A function in which there exists two or more than two different elements in domain having the same image in co-domain, and all the elements of co-domain must have at least one pre-image. A function is many-one onto if and only if it is both many-one and onto(surjective) functions.
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