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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Originally Posted by fifthrapiers

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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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.