Thread: 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.

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 so that no two pairs have the same difference?”

Originally Posted by Plato

This equivalent to asking “Is it possible to select five different elements from 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

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

That is exactly what I said.
Is that possible?

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?

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.