Prove of number of subsets of a set

[rajeshmarndi]

Homework Statement

A= {a₁,...a_m}, B= {b₁,...a_n}. If f: A→B is a function, then f(a₁) can take anyone of the n values b₁,...b_n. Similarly f(a₂). Then there are n^m such function. I understand this part.

So in my book, using this principle,
ⁿC₀ + ⁿC₁ + ... + ⁿC_n = 2ⁿ is proved.

It has taken a set A= {a₁,...a_n}. For each subset B of A, there is a function f_B : A→ {0,1} given by f_B(a₁) = 1 if a₁ ∈ B and f_B(a₁) = 0 otherwise. Conversely, for every f: A→ {0,1}, B = {a₁ : f_B(a₁) =1 } ⊂ A anf f = f_B. Thus there is a one-to-one correspondense between the subsets of A and the functions f: A→ {0,1}. From the above formula, hence we have
ⁿC₀ + ⁿC₁ + ... + ⁿC_n = 2ⁿ

Homework Equations

The Attempt at a Solution

[/B]The prove given is so short for me to understand anything.

 

[SammyS]

Homework Statement

A= {a₁,...a_m}, B= {b₁,...b_n}. If f: A→B is a function, then f(a₁) can take anyone of the n values b₁,...b_n. Similarly f(a₂). Then there are n^m such function. I understand this part.

So in my book, using this principle,
ⁿC₀ + ⁿC₁ + ... + ⁿC_n = 2ⁿ is proved.

It has taken a set A= {a₁,...a_n}. For each subset B of A, there is a function f_B : A→ {0,1} given by f_B(a₁) = 1 if a₁ ∈ B and f_B(a₁) = 0 otherwise. Conversely, for every f: A→ {0,1}, B = {a₁ : f_B(a₁) =1 } ⊂ A and f = f_B. Thus there is a one-to-one correspondense between the subsets of A and the functions f: A→ {0,1}. From the above formula, hence we have
ⁿC₀ + ⁿC₁ + ... + ⁿC_n = 2ⁿ

Homework Equations

The Attempt at a Solution

[/B]The prove given is so short for me to understand anything.

It may help to take the principle you are applying and restate it using other variables.

Let D= {d₁, ... , d_m}, R= {r₁, ... , r_n}. If f: D→R is a function, then f(d₁) can take anyone of the n values r₁, ... , r_n. Similarly for f(d₂). Then there are n^m such functions.​

Now apply this to the subset situation.

 

[rajeshmarndi]

Now apply this to the subset situation.

Sorry, that is the problem, I am unable to link the subset to the above mentioned situation.

 

[vela]

Sorry, that is the problem, I am unable to link the subset to the above mentioned situation.

What's stopping you? What part don't you understand? Simply saying "I don't get it" isn't very helpful.