Question

Let S be a subset of the set of irrational numbers such that the sum of any two distinct elements of S be a rational number. Find the maximum possible number of distinct elements in S.

Answer 1

Answer:

2

Step-by-step explanation:

1) We have,

$Q^c=Set\:of\:Irrationals\\ \\Q=Set\:of\:Rationals$

2) We need,

S be a subset of of the set of irrationals such that sum of any two distinct elements of S be a rational number.

That is,

$S=[x,y:x,y \in Q^c \:\&\:x+y\in Q]$

This is possible  when both x,y are such that x+y=0 .

3) Hence,

$S=[x,y:x,y \in Q^c \:\&\:x+y=0]\\ \\=[x,-x]$

That is,

Maximum number of distinct elements in S are 2.

(one is negative of other).