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Recall that the set {0,1}? is the set of all finite-length binary strings. Let f:{0,1}??{0,1}? to...

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Advance Math

  1. Recall that the set {0,1}? is the set of all finite-lengthbinary strings. Let f:{0,1}??{0,1}? to be f(x1x2…xk)=x2x3…xkx1.That is, f takes the first bit of a string x and moves it to theend of x, so for example a string 100becomes 001; if |x|?1, thenf(x)=x Also, suppose that g:{0,1}??{0,1}? is a function such thatg(x1…xk)=0x1…xk (that is, gg puts an extra 0 in front of the givenstring, so for example g(100)=0100. Everywhere in this question wewill refer to these f and g.
    1. Then  f(0011010)= and f(1)=
    2. Then g(000)=andg(1)=
    3. Then f?1(0011010)=
    4. Is f one-to-one? Is it onto? Is it a bijection?
    5. Is g one-to-one? Onto? Bijection?
    6. Calculate f(g(100101)) and g(f(100101)).
    7. Which of the following is true for these ff and gg? Justifyyour answer.
      1. ?x?{0,1}?,f(g(x))=g(f(x))
      2. ?x?{0,1}?,f(g(x))?g(f(x))
      3. Neither

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