1. ## Set problem

Hello I need help with this problem

Let
S be a set with n elements.

(a)How many ordered pairs are there in
S × S ?
(b) How many relations are there on
S (subsets of S × S) ?

Let
A, B, C be 3 sets. Suppose |A| = |B| = |C| = 100, |A \ B| = 70,

|
A \ C|
= 50, |B \ C| = 45 and |A [B [ C| = 175. How many elements are there in A \ B \ C ?

\ means intersection and [ means union

Thank you

2. Originally Posted by qwerty321
Hello I need help with this problem

Let
S be a set with n elements.

(a)How many ordered pairs are there in
S × S ?
(b) How many relations are there on
S (subsets of S × S) ?

Let
A, B, C be 3 sets. Suppose |A| = |B| = |C| = 100, |A \ B| = 70,

|
A \ C|
= 50, |B \ C| = 45 and |A [B [ C| = 175. How many elements are there in A \ B \ C ?

\ means intersection and [ means union

Thank you

(a) If $\displaystyle (x, y) \in S \times S$, since there are n elements in S, you have n choices for x and n choices for y. So $\displaystyle n*n = ?$
(c) Suggestion: Draw a Venn Diagram.

3. Originally Posted by qwerty321
Let [/FONT]A, B, C be 3 sets. Suppose |A| = |B| = |C| = 100, |A \ B| = 70,[/LEFT]
|
A \ C|
= 50, |B \ C| = 45 and |A [B [ C| = 175. How many elements are there in A \ B \ C ?

\ means intersection and [ means union

Thank you

Use this $\displaystyle \left| {A \cup B \cup C} \right| = \left| A \right| + \left| B \right| + \left| C \right| - \left| {A \cap B} \right| - \left| {A \cap C} \right| - \left| {B \cap C} \right| + \left| {A \cap B \cap C} \right|$

4. from where u got the formula?
thank you
and could you help me with the question for the relation?

5. Originally Posted by qwerty321
from where u got the formula?
That is known as inclusion/exclusion. It is one of the most important counting tools.

Originally Posted by qwerty321
(b) How many relations are there on S
$\displaystyle \begin{gathered} \left| S \right| = n \hfill \\ \left| {S \times S} \right| = n^2 \hfill \\ \left| {\mathbb{P}(S \times S)} \right| = 2^{n^2 } \hfill \\ \end{gathered}$