See Last Minute Notes on all subjects here.

## Relations And Functions

|A| = m and |B| = n , then

1. No. of functions from A to B = n^{m}

2. No. of one to one function = (n,P,m)

3. No. of onto function =nm – (n,C,1)*(n-1)^{m} + (n,C,2)*(n-2)^{m} …. +(-1)^{m}*(n,C,n-1)

4. Necessary condition for surjective function |A| = |B|

5. The no. of bijection function =n!

6. No. of relations =2^{mn}

7. No. of reflexive relations =2^{n(n-1)}

8. No. of symmetric relations = 2^{n(n+1)/2}

9. No. of Anti Symmetric Relations = 2^{n}*3^{n(n-1)/2}

10. No. of asymmetric relations = 3^{n(n-1)/2}

11. No. of irreflexive relations = 2^{n(n-1)}

12. A relation is partial order if

1) Reflexive 2) Antisymmetric 3) Transitive

13. Meet Semi Lattice :

For all a,b belongs to L a^b exists

14. Join Semi Lattice

For all a,b belongs to L aVb exists

15. Poset is called Lattice if it is both meet and join semi lattice

16. Complemented Lattice : Every element has complement

17. Distributive Lattice : Every Element has zero or 1 complement .

18. Boolean Lattice : It should be both complemented and distributive . Every element has exactly one complement.

## Graph Theory

1. No. of edges in a complete graph = n(n-1)/2

2. Bipartite Graph : There is no edges between any two vertices of same partition . In complete bipartite graph no. of edges =m*n

3. Sum of degree of all vertices is equal to twice the number of edges.

4. Maximum no. of connected components in graph with n vertices = n

5. Minimum number of connected components =

0 (null graph) 1 (not null graph)

6. Minimum no. of edges to have connected graph with n vertices = n-1

7. To guarantee that a graph with n vertices is connected , minimum no. of edges required = {(n-1)*(n-2)/2 } + 1

8. A graph is euler graph if it there exists atmost 2 vertices of odd – degree

9. Tree

-> Has exactly one path btw any two vertices -> not contain cycle -> connected -> no. of edges = n -1

10. For complete graph the no . of spanning tree possible = n^{n-2}

11. For simple connected planar graph

1). ∑d(Ri) = 2|E| 2). |V| + |R| = |E| + 2 3). 3|R| <= 2|E| 4). |E| <= 3|v| - 6

12.) Every bipartite graph is 2 colourable and vice versa

13.) The no. of perfect matchings for a complete graph (2n)/(2^{n}n!)

14.) The no. of complete matchings for K_{n.n} = n!