Please do all parts.

3. (10 points) Let E be the set of 4-bit binary strings: E = {0000, 0001, 0010, ... , 1111}. (a) Let R be the following relation in E: rRy if r and y have the same number of 1's. Prove that R is an equivalence relation, and give the equivalence classes of R. (b) The Hamming distance between two strings r and y in E, denoted H(r,y), is the number of bit- wise difference between r and y, that is, the minimum number of bits that have to be flipped in the first string to become identical to the second string. For example, H (0000, 0101) = 2, H (1101, 0110) = 3, and H(1001, 0110) = 4. Let R be the following relation in E: Ry if H(1,y) is even. Prove that R is an equivalence relation, and give the equivalence classes of R. = = 3. (10 points) Let E be the set of 4-bit binary strings: E = {0000, 0001, 0010, ... , 1111}. (a) Let R be the following relation in E: rRy if r and y have the same number of 1's. Prove that R is an equivalence relation, and give the equivalence classes of R. (b) The Hamming distance between two strings r and y in E, denoted H(r,y), is the number of bit- wise difference between r and y, that is, the minimum number of bits that have to be flipped in the first string to become identical to the second string. For example, H (0000, 0101) = 2, H (1101, 0110) = 3, and H(1001, 0110) = 4. Let R be the following relation in E: Ry if H(1,y) is even. Prove that R is an equivalence relation, and give the equivalence classes of R. = =