Question

If set S has |S| = 3 elements and satisfies
i. 0 ∈ S and
ii. if p ∈ S, q ∈ S and p ≠q , then (p+q) ∈ S
and 10 is an element of S, what is the other element of S excluding 0 and 10.

Answer 1

Hi friend!!

Given, If set S has |S| = 3 elements and satisfies
i. 0 ∈ S

which says that '0' is an element of S.

Then it says, ii. if p ∈ S, q ∈ S and p ≠q , then (p+q) ∈ S
and 10 is an element of S. let p =10

so, we came to know that 0,10,q are the elements of S .

it also says, (p+q)€S

which indirectly says, p+q=0

10+q=0

q=-10.

I hope this will help u;)

Answer 2

$\text{Let S be the set}\\S=\{0,10,a\} ~~~(a \neq 0,a\neq 10)\\Since (10+a)\in S \text{ from (ii) we have,}\\10+a = 0 \ or \ 10+a = 10,\\which \ implies \ a = -10 \ or \ a = 0\\\\\text{Since, } |S| = 3, this\ implies,\\S = {0,10,-10}\\\\Hence,\text{other element of S excluding 0 and 10 is -10}.$