# Last Minute Notes – Engineering Mathematics

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 = nm
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 =2mn
7. No. of reflexive relations =2n(n-1)
8. No. of symmetric relations = 2n(n+1)/2
9. No. of Anti Symmetric Relations = 2n*3n(n-1)/2
10. No. of asymmetric relations = 3n(n-1)/2
11. No. of irreflexive relations = 2n(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 = nn-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)/(2nn!)
14.) The no. of complete matchings for Kn.n = n!

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