# Math Help - Prove about no bijective fuction exist

1. ## Prove about no bijective fuction exist

Let S={a,b,c,d,e} and let T be the set of all ten 2-element subsets of S. Show that there exists no injective function f:S→{0,1,2,...,|T|} such that the function g: T→ {1,2,...,|T|} defined by g({i,j})=|f(i)-f(j)|is bijective.

2. Originally Posted by zhushp
Let S={a,b,c,d,e} and let T be the set of all ten 2-element subsets of S. Show that there exists no injective function f:S→{0,1,2,...,|T|} such that the function g: T→ {1,2,...,|T|} defined by g({i,j})=|f(i)-f(j)|is bijective.
This equivalent to asking “Is it possible to select five different elements from $\{0,1,\cdots,9,10\}$ so that no two pairs have the same difference?”

3. Originally Posted by Plato
This equivalent to asking “Is it possible to select five different elements from $\{0,1,\cdots,9,10\}$ so that no two pairs have the same difference?”
not exactly, it is asking to proof that there're no two pairs have the same difference from every five different elements in $\{0,1,\cdots,9,10\}$

4. Originally Posted by zhushp
not exactly, it is asking to proof that there're no two pairs have the same difference from every five different elements in $\{0,1,\cdots,9,10\}$
That is exactly what I said.
Is that possible?

5. Originally Posted by Plato
That is exactly what I said.
Is that possible?
i think is impossible since i cannot come out any counter-example, but how to proof?

6. Originally Posted by Plato
That is exactly what I said.
Is that possible?
Not exactly. You did not specify that by "difference" between a and b you mean |a- b|, not a-b or b-a.