Chapter 7: Q.7.16 (page 360)

Suppose that each of the elements of $S = {1, 2, \dots, n}$ is to be colored either red or blue. Show that if $A_{1}, \dots, A_{r}$ are subsets of $S$ , there is a way of doing the coloring so that at most $\sum_{i = 1}^{r} (1 / 2)^{|A_{i}| - 1}$ of these subsets have all their elements the same color (where $| A |$ denotes the number of elements in the set $A$ ).

Short Answer

Expert verified

It has been shown that at most $\sum_{i = 1}^{r} {(\frac{1}{2})}^{[A_{i}] - 1}$ of the subsets $A_{1}, A_{2}, \dots, A_{r} .$

Step by step solution

Given Information

Elements $S = {1, 2, . ., n}$ is to be colored either red or blue.

$A_{1}, \dots, A_{r}$ are subsets of $S$ ,

Show that at most $\sum_{i = 1}^{r} (1 / 2)^{|A_{i}| - 1}$ of the given subsets.

Explanation

Each of the element of $S = {1, 2, \dots, n}$ is to be colored red or blue suppose that each element is independently, equally likely to be colored Red or Blue.

Each elements is Red or Blue with probability $\frac{1}{2}$

Let $E_{i} = \{All the elements of ith subject A_{i} are of the same colour\}$

$∴ P [E_{i}] = 2 . {(\frac{1}{2})}^{[A_{i}]} = {(\frac{1}{2})}^{[A_{i}] - 1} \forall i = 1, 2, . ., n$

Where $[A_{i}] =$ Number of elements in $A_{i} \forall i = 1, 2, . ., n$

$∴ \sum_{i = 1}^{r} P [E_{i}] = \sum_{i = 1}^{r} {(\frac{1}{2})}^{[A_{i}] - 1}$

Explanation

As from Boole's inequality

$P [\underset{i}{\cup} E_{i}] \leq \sum_{i} P [E_{i}]$

$\Rightarrow$ At most $\sum_{i = 1}^{r} {(\frac{1}{2})}^{[λ λ_{i}] - 1}$ of the subsets of $A_{1}, A_{2}, \dots, A_{r}$

Final Answer

Hence, it has been shown that at most $\sum_{i = 1}^{r} (1 / 2)^{|A_{i}| - 1}$ of the subset $A_{1}, A_{2}, \dots, A_{r} .$

And have all their elements the same color.

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