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Prove that the n-cube is connected, for n ≥ 1

  1. Mar 23, 2007 #1
    hello

    any pone have any idea how to solve this question

    thanx

    Prove that the n-cube is connected, for n ≥ 1.
    (Hint: build the n-cube using two copies of (n − 1)-cube and use induction
    on n.)
     
  2. jcsd
  3. Mar 24, 2007 #2

    HallsofIvy

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    Think about what they are asking you to do. The "0-cube" is a single point. The "1-cube" is a line segment. You "build the 1-cube using two copies of the 0-cube" by connecting two points. If one of the points is labled "0" and the other is labled "1" then every point on the 1-cube is labled "x" for 0<= x<= 1. I assume that you are allowed to use the fact that the interval [0, 1] is connected.

    The 2-cube is a square. You "build the 2-cube using two copies of the 1-cube" by using each one cube as an edge. In particular, if every point on one 1-cube is labled (x, 0) and the other (x, 1), then every point in the 2-cube is labled (x,y) with 0<= y< = 1.

    If every point on one n-1-cube is labled (x1, x2, ..., xn-1, 0) and every point on the other n-1-cube is labled (x1, x2, ..., xn-1, 1) then every point on the n-cube is labled (x1, x2, ..., xn-1, xn[/sup]). Now use the fact that the n-1-cube and the interval [0,1] is connected to prove that the n-cube is connected.
     
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