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) ].