The document defines key concepts related to functions including: - A function maps elements of one set (the domain) to elements of another set (the range). - Composite functions can be formed by applying multiple functions sequentially. - A function is one-to-one if each element in the range has at most one element in the domain mapping to it. A function is onto if each element in the range has at least one element in the domain mapping to it. - Permutations are bijective (one-to-one and onto) functions from a set to itself. Permutations can be represented as products of cycles or transpositions.

2. Function
 A function or mapping ∅ from a set A into a set B is a
rule which assigns to each element a of A exactly one
element b of B. We say that ∅ maps a into b, and that ∅
maps A into B.

3. Notations
 Classical notations
“∅ 𝑎 = 𝑏”
read as “∅ maps a into b” or “the image of a under ∅ is
b”
 Present day trend among algebraists. The notation
“a∅ = 𝑏”
 The fact that ∅ maps A into B will be symbolically
expressed by
“∅: 𝐴 → 𝐵”

4. Composite Function
 If ∅ and 𝜔 are functions with ∅: 𝐴 → 𝐵 and 𝜔: 𝐵 → 𝐶,
then there is a natural function mapping A into C.
That is, you can get from A to C via B, using the
function ∅ and 𝜔. This function mapping A into C is
the composite function consisting of ∅ followed by 𝜔.
 Classical notation
∅ 𝑎 = 𝑏 and 𝜔 𝑏 = 𝑐, so 𝜔 ∅ 𝑎 = 𝑐
 Present day
𝑎 ∅𝜔 = 𝑐

5. Definition
 A function from a set A into a set B is one to one if each
element of B has at most one element of A mapped
into it, and is onto B if each element of B has at least
one element of A mapped into it.
1. to show that ∅ is one to one you show that ∅ 𝑎1 =
∅ 𝑎2 implies that 𝑎1 = 𝑎2.
2. To show that ∅ is onto B, you show that for each 𝑏 ∈
𝐵, there exists 𝑎 ∈ 𝐴 such that ∅ 𝑎 = 𝑏.

6. Domain and Range
 ∅: 𝐴 → B. A is the domain of ∅, and if ∅ is onto B,
then B is the range of ∅ or the image of A under ∅.

7. Example
1. Consider ∅ = 𝑓 𝑥 , where “∅: 𝑅 → R”. Show that
𝑓 𝑥 = 𝑥2 + 2 is a one to one function.
2. Consider ∅ = 𝑓 𝑥 , where “∅: 𝑅 → R+
”. Show that
𝑓 𝑥 = 𝑒𝑥
is a onto function.
3. Consider ∅ = 𝑓 𝑥 , where “∅: 𝑅 → R”. Show that
𝑓 𝑥 = 𝑥3
+ 3 is a one to one function

8.  Consider ∅ = 𝑓 𝑥 , where “∅: 𝑅 → R+
”. Show that
𝑓 𝑥 = 𝑒𝑥
is a onto function.

9. Permutation
 A permutation of a set is the rearrangement of the
elements of the set.
 A permutation of a set A is a function from A into A
which is both one to one and onto. In other words, a
permutation of A is a one-to-one function from A onto
A.

10. Example
1 4
2 2
3 5
4 3
5 1
1 3
2 2
3 4
4 5
5 3
Figure 1.2 Figure 1.3

11. Permutation Multiplication
 Permutation multiplication is defined on the
permutations of a set. Let A be a set, and let ∅ and 𝜏 be
permutations of A so that ∅ and 𝜏 are both one – to –
one function mapping A onto A.
 The composite function ∅𝜏 gives a mapping of A into
A. Now ∅𝜏 will be a permutation if it is one to one and
onto A.
 The permutation 𝑖 such that 𝑖 𝑎 = 𝑎 for all 𝑎 ∈ 𝐴 acts
as identity.
𝑖 =
1 2
1 2
⋯
⋯
𝑛
𝑛

12. Example
 If ∅ =
1 2 3
4 2 5
4 5
3 1
and 𝜏 =
1 2 3
3 5 4
4 5
2 1
,
then
(a) ∅𝜏=
1 2 3
4 2 5
4 5
3 1
1 2 3
3 5 4
4 5
2 1
=
1 2 3
5 1 3
4 5
2 4
(b) 𝜏∅ =
1 2 3
3 5 4
4 5
2 1
1 2 3
4 2 5
4 5
3 1
=
1 2 3
2 5 1
4 5
4 3

13. ∅ =
1 2 3
4 2 5
4 5
3 1
c. ∅′ =
1 2 3
5 2 4
4 5
1 3
d. ∅′𝜏=
1 2 3
5 2 4
4 5
1 3
1 2 3
3 5 4
4 5
2 1
=
1 2 3
4 3 1
4 5
2 5
e. 𝜏∅′𝜏=
1 2 3
3 5 4
4 5
2 1
1 2 3
5 2 4
4 5
1 3
1 2 3
3 5 4
4 5
2 1
=
1 2 3
2 4 3
4 5
5 1
f. 𝜏2
=
1 2 3
3 5 4
4 5
2 1
1 2 3
3 5 4
4 5
2 1
=
1 2 3
4 1 2
4 5
5 3

14. Definition (Orbits of 𝜎)
 The orbit of an element x∈X is apparently simply the
set of points in the cycle containing x.
 Example 1. Find the orbits of the permutation
𝜎 =
1 2 3
3 8 6
4 5 6
5 7 1
7 8
4 2
The complete list of orbits of 𝜎 is
{1,3,6}, {2,8}, {4,5,7}

15. Definition (Cycle)
 A permutation 𝜎 ∈ 𝑆𝑛 is a cycle if it has at most one
orbit containing more than one element. The length of
a cycle is the number of elements in its largest orbit.
Example:
𝜎 =
1 2 3
3 2 6
4 5 6
4 5 1
7 8
7 8
=(1,3,6)

16. Theorem
 Every permutation 𝜎 of a finite set is a product of
disjoint cycles.
 Example:
Consider the permutation
1 2 3
6 5 2
4 5 6
4 3 1
= 1, 6 2, 5, 3

17. Example 2
 Let 1, 4, 5, 6 and 2, 1, 5 be cycle in 𝑆6. Find
(a) 1, 4, 5, 6 2, 1, 5
(b) 2, 1, 5 1, 4, 5, 6
1 2 3
4 2 3
4 5 6
5 6 1
1 2 3
5 1 3
4 5 6
4 2 6
1 2 3
6 4 3
4 5 6
5 2 1
=(1,6)(2,4,5)

19. Definition (Transposition)
 A cycle of length 2 is a transposition.
Corollary
Any permutation of a finite set of at least two elements is
a product of transposition.
𝑎1, 𝑎2, ⋯ , 𝑎𝑛 = 𝑎1, 𝑎𝑛 𝑎1, 𝑎𝑛−1 ⋯ 𝑎1, 𝑎3 𝑎1, 𝑎2
Example:
1,6 2,5,3 = 1,6 2,3 2,5

20. Definition (Even or odd
permutation)
 A permutation of a finite set is even or odd according
to whether it can be expressed as a product of an even
number of transpositions or the product of an odd
number of transpositions, respectively.
 Example.
1. 2,4,6,8 =(2,8)(2,6)(2,4) Odd permutation
2. (1,3,2,5,6)=(1,6)(1,5)(1,2)(1,3) even permutation

21. Exercises 2
a. Write the permutations (23) and (13)(245) on 5
symbols in two line notation.
b. Express the products (23) 0 (13)(245) and (13)(245) 0
(23) in cyclic notation.
c. Express in cyclic notation the inverses of (23) and of
(13)(245).
2. Show that the products of the cycles (1357) and (2468)
are commutative. State the theorem covering this.

22. Express the products (23) 0 (13)(245) and
(13)(245) 0 (23) in cyclic notation.

23. Express in cyclic notation
the inverses of (23) and of
(13)(245).

24. Show that the products of the cycles (1357)
and (2468) are commutative. State the
theorem covering this.

25. Assignment
1. Which of the following function is one-to-one or onto
or both one-to-one and onto functions . Such that ∅ =
𝑓 𝑥 , where “∅: 𝑅 → 𝑅+
”
a. 𝑓 𝑥 = 2𝑥 + 3
b. 𝑓 𝑥 = 3𝑥3
+ 2𝑥2
− 2𝑥 +5
c. 𝑓 𝑥 = 2𝑥2
− 5𝑥 + 3

26. Exercises
1. Express each of the permutations on 8 symbols as a product
of disjoint cycles and as a product of transposition.
(a)
1 2 3
2 3 4
4 5 6
1 5 6
7 8
7 8
(b)
1 2 3
3 4 5
4 5 6
6 7 8
7 8
1 2
(c)
1 2 3
3 4 1
4 5 6
6 8 2
7 8
7 5
(d) 2,4,6,8 ° 3,4,8
(e) 1,5 2,4,6,8 ° 3,7 1,5,4,6,8
(f) 1,3,5 ° 3,4,5,6 ° 4,6,7,8