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Math Help - Equiv. Relations + Proof!

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    Equiv. Relations + Proof!

    1.) Let R be the relation on R x R given by (x,y) R (z,w) if and only if y = w.
    (a) Prove that R is an equivalence relation on R x R.

    (b) Give a geometric description of the equivalence class that contains the pair (1,1).

    (c) We learned a theorem that states that an equivalence relation partitions the underlying set. In this particular case, give a geometric description of the partition of R x R that results from the equivalence relation R.

    2.) Give a complete proof that the number of 0-1 sequences of length n 4 that contain exactly two occurrences of 01 is given by (n + 1) choose (5) [or C(n+1,5) ].
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    Quote Originally Posted by fifthrapiers View Post
    1.) Let R be the relation on R x R given by (x,y) R (z,w) if and only if y = w.
    (a) Prove that R is an equivalence relation on R x R.
    (b) Give a geometric description of the equivalence class that contains the pair (1,1).
    (c) We learned a theorem that states that an equivalence relation partitions the underlying set. In this particular case, give a geometric description of the partition of R x R that results from the equivalence relation R.

    2.) Give a complete proof that the number of 0-1 sequences of length n 4 that contain exactly two occurrences of 01 is given by (n + 1) choose (5) [or C(n+1,5) ].
    Surely you can do part a! It is a simple as knowing that equality is an equivalence relation.
    For part b, think horizontal line. Then for part c, do horizontal lines intersect?

    I simply cannot understand what you have written for #2.
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