Sets, Relations, functions Flashcards
No of elements in cartesian product
m * n
No of relations
2^mn
For f(g(x)) to be onto and one one
f(x) must be onto
g(x) must be one one
No of reflexive relations
2^(n^2 -n)
No of symmetric relations
2^(n(n+1)/2)
How to find max of (ax^2 + bx +c)/px^2 + qx + r)
Let it be equal to m
Then put D>= 0
no of one one functions from set containing m to set containing n elements
nPm
no of onto functions from set containing m elements to set containing n elements
sum r=1 to n nCr r^m (-1)^n-r
Period of sinx cosx
2pi
time period of secx cosecx
2pi
time period of tanx cotx
pi
time period of square of any trig fn
pi
time period of mod of any trig fn
pi
Period of fractional part x
1
Solution for f(x) = f^-1(x)
f(x) = x
Period of f(x) is P, then period of f(ax+b) is
P/mod(a)
inequality between x and e^x
e^x > x always
converting cot^-1 to tan^-1 , both the cases
if x is positive, cot^-1(x) = tan^-1(1/x)
else, cot^-1(x) = pi + tan^-1(1/x)
sin^-1x + sin^-1y
sin^-1(xroot(1-y^2) + yroot(1-x^2))
cos^-1x + cos^-1y
cos^-1(xy - root(1-y^2)root(1-x^2))
