1

I want to find the example of $$f\left(\bigcap\limits_{i\in I}A_i\right)\subset \bigcap\limits_{i\in I}f\left(A_i\right).$$(proper subset).

I have find the example of that, but $$f\left(\bigcap\limits_{i\in I}A_i\right)= \bigcap\limits_{i\in I}f\left(A_i\right),$$ like this

\begin{eqnarray} f:\mathbb{Z}&\to& \mathbb{Z}\newline x&\mapsto& x^2 \end{eqnarray}

$A_1=\{2,3,5,7,11\}$

$A_2=\{1,2,3,4,5,6\}$

$A_3=\{1,3,5,7,9,11,13\}$

$\bigcap\limits_{i=1}^3A_i=\{3,5\}$

$f\left(\bigcap\limits_{i=1}^3A_i\right)=\{9,25\}$

$f(A_1)=\{4,9,25,49,121\}$

$f(A_2)=\{1,4,9,16,25,36\}$

$f(A_3)=\{1,9,25,49,81,121,169\}$

$\bigcap\limits_{i=1}^3 f(A_i)=\{9,25\}$

So,

$\bigcap\limits_{i=1}^3 f(A_i) = f\left(\bigcap\limits_{i=1}^3A_i\right)$.

Now I want to find the example of $$f\left(\bigcap\limits_{i\in I}A_i\right)\subset \bigcap\limits_{i\in I}f\left(A_i\right).$$(proper subset).

Can anyone help me to find it?

Asaf Karagila
  • 393,674
Ongky Denny Wijaya
  • 2,298

1 Answers1

2

$f(x)=x^2,A_1=\{1,2\},A_2=\{1,-2\}$

Shubham Johri
  • 17,659
  • Yes, your answer is right.. $A_1\cap A_2 = {1}$, $f(A_1\cap A_2)={1}$, $f(A_1)={1,4}$, $f(A_2)={1,4}$, so $f(A_1)\cap f(A_2)={1,4}$. So, $f(A_1\cap A_2)\subset f(A_1)\cap f(A_2)$. Thank you for your help. – Ongky Denny Wijaya Feb 06 '19 at 07:35