(4) Let X = {(a,b) : a,bЄ Z, a on X given by (a, b) R(c, d) if ad 0,60}. Show that the relation R = bc is an equivalence relation. [ 1 for reflexivity, 1 for symmetry, 1 for transitivity, 1 for conclusion]

In each of the following parts, a relation is defined on the set of all human beings. Determine whether the relation is reflective, symmetric, or transitive. Justify your answers. xRy if and only if x lives within 400 miles of y. xRy if and only if x is the father of y. xRy if and only if x is a first cousin of y. xRy if and only if x and y were born in the same year. xRy if and only if x and y have the same mother. xRy if and only if x and y have the same hair colour.

2) Let Q = {0,1,2,3} and define relation R on Q as follows: R = {(0,0), (0,1), (0,3), (1,0), (1,1), (2,2), (3,0), (3,3)} Determine whether the R relation is reflexive, symmetric or transitive.

Let A = {1, 2, 3, 4, 5} and R be the relation defined by R = {(1,1), (2,2), (2,4), (2,5), (3,3),(4,2), (4,4)}. Justify whether relation R fulfill the property of: i) Reflexive. ii) Symmetric.(iii) Anti-symmetric.(iv) Transitive.

Let 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.

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I need help with this problem involving equivilence relations

3. For universe S = R2 define relation ~ as: (a, b) ~ (c,d) — 2a-b2c - d. Prove that is an equivalence relation on R2, or find a counterexample.

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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)