Question-216734

Question Number 216734 by mnjuly1970 last updated on 17/Feb/25

Answered by sniper237 last updated on 18/Feb/25

L e t E = {f : A \to B}, ∣ E ∣= 3^{m}

E ∖ S = {f \in E, ∣ f (A) ∣= 1 o r 2}

∣ E ∖ S ∣= C_{3}^{1} . 1^{m} + C_{3}^{2} . (2^{m} - 1^{m} - 1^{m}) = 3 . 2^{m} - 3

S o ∣ S ∣= 3^{m} - (3 . 2^{m} - 3)

Commented by mehdee7396 last updated on 18/Feb/25

n o t e : 3^{m} - (3 \times 2^{m} - 3) \neq 3^{m} - 3 \times 2^{m} - 3

Commented by mnjuly1970 last updated on 18/Feb/25

t h a n k s a l o t \dots

Answered by mehdee7396 last updated on 19/Feb/25

l e t A = {a_{i} : 1 ⩽ i ⩽ m} & B = {b_{1}, b_{2}, b_{3}}

F = {f : A \to B ∣ f (a) = b; a \in A, b \in B}

f_{1} = {f : A \to B ∣ f (a_{i}) \neq b_{1}}

f_{2} = {f : A \to B ∣ f (a_{2}) \neq b_{2}}

f_{3} = {f : A \to B ∣ f (a_{i}) \neq b_{3}}

\Rightarrow∣ F ∣= 3^{m} & ∣ f_{i} ∣= 2^{m} & ∣ f_{i} \cap f_{j} ∣= 1 & ∣ f_{1} \cap f_{2} \cap f_{3} ∣= 0

\Rightarrow∣ S ∣=∣ F ∣ - Σ ∣ f_{i} ∣ + Σ ∣ f_{i} \cap f_{j} ∣ - ∣ f_{1} \cap f_{2} \cap f_{3} ∣

= 3^{m} - 3 \times 2^{m} + 3