Question

Give partimos of R having (a) one block, (b) two blocks, (c) three blocks, (d) infinity many blocks

Answer 1

(a) If we include the entire real space in our set ; S=R; then we have made 1 partition of R.

(b) If we partition the numbers in real space as rational or irrational, i.e. $R= \text\textbraceleft Q, Q' \text\textbraceright$ , where

Q= set of all rational numbers and

Q'=set of all irrational numbers;

then we have made 2 partitions of R.

(c)If we partition the numbers in real space on the basis of their signs , then $R= \text\textbraceleft P,N, Z \text\textbraceright$; where

P=set of all positive numbers in R.

N= set of all negative numbers in R.

Z={0} ---(the only unsigned number in R);

then we have made 3 partitions in R.

(d) Let $R= \text\textbraceleft R= \text\textbraceleft 1 \text\textbraceright,\text\textbraceleft 2 \text\textbraceright,\text\textbraceleft 3\text\textbraceright ,...\text\textbraceright$

If we consider the above set in which R is composed only of singletons, where each singleton represents a unique number in R, then we have made infinitely many partitions in R.