7. Define a relation p on N × N by (a, b)p(c, d) if and only if there is ad = bc. (a) Show that p is an equivalence relation. (b) As always, we use [(2, 3)] to mean the equivalence class of (2, 3). Determine if (1,2) = [(2,3)], (4,6) = [(2,3)] and (10,15) = [(2,3)].
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- Let and be lines in a plane. Decide in each case whether or not is an equivalence relation, and justify your decisions. if and only ifand are parallel. if and only ifand are perpendicular.Q2. (a) Find the equivalence class [-3]5 (b) Find the ordered pairs of the relation R from the set A = {0, 1,2, 3} into the set B = {2,3,5}, defined by R = {(x,y):x + 3 < y}4. Let R be the relation {(1, 2), (1, 3), (2, 3), (2, 4), (3, 1)), and let S be the relation {(2, 1), (3, 1), (3, 2), (4, 2)}. Find S.R.
- The real line R is divided into subsets X1, X2, X3 where X1 = (-00, – 7], X2 = [-7,1), and X3 = [1, 00). Can X1, X2, X3 be equivalence classes with respect to some equivalence relation on R? O No, they can't be equivalence classes for some equivalence relation since X1n X2 + 0. O No, they can't be equivalence classes for some equivalence relation since any equivalence relation on infinite set has infinitely many different equivalence classes. O Yes, these sets can be equivalence classes for some equivalence relation since X1 U X2 U X3 = R. O Yes, these sets can be equivalence classes for some equivalence relation since X2n X3 = 0.For each relation on {0, 1, 2, 3} below show that the relation is an equivalence relation, or show which qualities of an equivalence relation it lacks: {(0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3)} { (0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 2), (3, 3)}Please help me with the below question.
- ii. Determine whether the relation, R = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)} is: Yes/No Justify your answer. Types of relations Reflexive Symmetric Antisymmetric Transitive (8 marks)Plz answer all parts correctly asapLet x1, x2, y1, y2 be real numbers. The expression (x1, y1) ~ (x2, y2) means that x12 + y12 = x22 + y22. Prove that the relation ~ is an equivalence relation on R2.
- 7. Let (a, b), (c, d) E R2. Define a relation ~on R² by (a, b) (c,d) if 2a - b = 2c - d. (a) Show that is an equivalence relation. (b) Find the equivalence class of (0, 1).The R is the relation from {1, 2, 3} to {1, 2, 3, 4} with R = {(1, 1), (1, 2), (2, 3),(3, 4), (3, 4)} and S is the relation from {1, 2, 3, 4} to {0, 1, 2} with S = {(1, 0),(2, 1), (3, 2), (3, 0), (4, 1)}. Find the composites of the relations SoR and RoS.R={(0,4), (3,0), (3,1), (4,2)}. Find transitive closer of R. Show diagram. A=Z+ x Z+ show relation R on A as for any (a,b) and (c,d), (a,b)R(c,d) (ab)=(cd) Show if R equ relation and explain List 4 elements in the equ class [(3,1)] Show distinct equ classes for R