M is the relation defined on Z as follows: a M b ⇐⇒ a ≡ b mod 5. (1) Prove that M is an equivalence relation. (2) Find the equivalence classe

Let T be the set {w = {0, 1}* ||w| ≤ 4}. Let R be the equivalence relation defined on T as follows: R = {(x, y) | x ≤T, yɛT, no(x) = = no(y)}, where no(r) represents the number of zeroes in the string x, and no(y) represents the number of zeroes in the string y. For example, (1011, 01) is a pair in R because the two strings 1011 and 01 have the same number of zeroes as each other. Every element in the set will appear in exactly one equivalence class and will be related to all elements in its class and not related to any elements outside of its class. What are the equivalence classes of T created by the relation R?

Let R3 be the relation on Z+ such that xR3y if and only if 2x – 3y > 0. (Recall that Z+ = {1,2,3, 4, ...}, the positive integers.) 14) Prove or Disprove: R3 is anti-symmetric.

Let R be the relation on Z defined by x Ry if and only if x+ 3y is even. Prove that R is an equivalence relation.

SOLVE STEP BY STEP IN DIGITAL FORMAT PLEASE Through the point P(-2,-7) tangents are drawn to the ellipse 2x² + y² + 2x-3y - 2 = 0; determine the equations of the tangents and the points of tangency +