Respuesta :
Answer:
Lets denote c the concatenation of strings. For a binary string a in B9, we define the element f(a) in E10 this way:
- f(a) = a c {1} if a has an odd number of 1's
- f(a) = a c {0} if a has an even number of 1's
Step-by-step explanation:
To show that the function f defined above is a bijective function, we need to prove that f is well defined, injective and surjective.
f is well defined:
To see this, we need to show that f sends elements fromo b9 to elements of E10. first note that f(a) has 1 more binary integer than a, thus, it has 10. if a has an even number of 1's, then f(a) also has an even number because a 0 was added. On the other hand, if a has an odd number of 1's, then f(a) has one more 1, as a consecuence it will have an even number of 1's. This shows that, independently of the case, f(a) is an element of E10. Thus, f is well defined.
f is injective (or one on one):
If a and b are 2 different binary strings, then f(a) and f(b) will also be different because the first 9 elements of f(a) form a and the first elements of f(b) form b, thus f(a) is different from f(b). This proves that f in injective.
f is surjective:
Let y be an element of E10, Let x be the first 9 elements of y, then f(x) = y:
- If x has an even number of 1's, then the last digit of y has to be 0, and f(x) = x c {0} = y
- If x has an odd number of 1's, then the last digit of y has to be a 1, otherwise it wont be an element of E10, and f(x) = x c {1} = y
This shows that f is well defined from B9 to E10, injective, and surjective, thus it is a bijection.
