Discrete Math Cheat Sheet

Probab­ility
1. P(x) = ⁿCx . p^x . (1-p)^n-x
2. P(x) = (^XCk)((^N-X)C(n-k))/^NCn

Set Theory
A = B when A subset of B & B subset of A
A - B = A n B'
A u (A n B) = A
A n (A u B) = A
A u A' = U
A n A' = nullset or {}

Power set of S is the set of ALL SUBSETS of S e.g. S = {1,2} , P(S) = { {}, {1}, {2}, {1,2}}

|A| = n, |P(A)| = 2ⁿ

Sets A and B are disjoint iff A n B = {}

Cardin­ality of union: |A u B| = |A| + |B| - |A n B|

Proof by induction:
Show that when p(k) is true, p(k + 1) follows.

A^T: Transpose of A - Switch Rows with Columns (R1 becomes C1, R2 becomes C2 etc.)

-A = -1 . A

A^-1: Inverse of A
A^-1 . I = I = A . I
A^-1A=I

Augment Identity matrix to matrix and perform Guass-­Jordon elimin­ation on both to get change Identity matrix to the Inverse.

EROs:
Switch Rows
Scale Row (Multiply entire row)
Add multiple of different row to another

A matrix A is in row echelon form if
1. The nonzero rows in A lie above all zero rows (when there is at least a nonzero row and a zero row).
2. The first nonzero entry in a nonzero row (called a pivot) lies to the right of the pivot in the row immedi­ately above it.