What are the key topics discussed in the class 11 mathematics Sets?

n P (A) = 2⁷ = 128

f : A → B

Number of function = 128 × 128….128 = 128⁷

$={\left(2^7\right)}^7=2^{49}$

->mⁿ = 2⁴⁹

m + n = 49 + 2 = 51

n (C) = 16, n (P) = 20, n (M) = 25

n (MÇP) = n (MÇC) = 15, n (PÇC) = 10,

n (MÇCÇP) = x.

n (CÈPÈM) £ n (U) = 40

n (CÈPÈM) = n (C) + n (P) + n (M) – n (C? M) – n (PÈM) – n (CÇP) + n (CÇPÇM)

40 ³ 16 + 20 + 25 – 15 – 15 – 10 + x

40 ³ 61 – 40 + x

19 ³ x

So maximum number of students that passed all the exams is 19.

$\sqrt[]{3}-\sqrt[]{2}=\frac{\left(\sqrt[]{3}+\sqrt[]{2}\right)\left(\sqrt[]{3}-\sqrt[]{2}\right)}{\left(\sqrt[]{3}+\sqrt[]{2}\right)}=\frac{1}{\sqrt[]{3}+\sqrt[]{2}}$

Let $\sqrt[]{3}+\sqrt[]{2}=t$  

$t^x+\frac{1}{t^x}=\frac{10}{3}$

Let  $t^x=y$ $y+\frac{1}{y}=\frac{10}{3}$           

y = 3 or   $\frac{1}{3}$

${\left(\sqrt[]{3}+\sqrt[]{2}\right)}^x=3$ or $\frac{1}{3}$  

$xlog\left(\sqrt[]{3}+\sqrt[]{2}\right)=ln3$ or –ln3

$x=\frac{ln3}{ln\left(\sqrt[]{3}+\sqrt[]{3}\right)}$   or $\frac{-ln3}{\sqrt[]{3}+\sqrt[]{2}}$  

two real values of x

$f\left(x\right)=\{\begin{array}{cc}\hfill e^{-x},\hfill & \hfill x<0\hfill \\ \hfill lnx,\hfill & \hfill x>0\hfill \end{array}$

$g\left(x\right)=\{\begin{array}{cc}\hfill e^x,\hfill & \hfill x<0\hfill \\ \hfill x,\hfill & \hfill x>0\hfill \end{array}$

The different types of sets covered in this chapter are - Empty Set, Singleton Set, Finite Set, Infinite Set, Equal Sets, Equivalent Sets, Subset, Proper Subset, Universal Set, Power Set, and Disjoint Sets.

Union (? ) in sets means "or". It gives a set of all unique elements of two or more sets. Intersection (? ) means "and". It includes all the common elements of two or more sets.