Any and all help greatly appreciated! For set S = {1, 2, 3, 4}, we define the following relationsR1 = {(2,2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)},R2 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)},R3 ={(2, 4), (4, 2)},R4 = {(1, 2), (2, 3), (3, 4)},R5 = {(1, 1), (2, 2), (3, 3), (4, 4)},Re = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)},Determine which of these statements are correct. Check ALL correct answers below.ORĮ is reflexiveR3 is transitiveO RĄ is antisymmetric| R3 is symmetric| R2 is reflexiveO RĄ is transitiveR3 is reflexiveR5 is not reflexiveO R6 is symmetricR5 is transitive|R2 is not transitiveR1 is not symmetricO RĄ is symetric In this problem we work out step-by-step the procedure for checking an equivalence relation. Denote by Z the setof all integers. Declare that two integers æ, y are related if x+y is an integer multiple of 7. In symbols:x ~ y + 7 divides x + yWe want to check if this is an equivalence relation. That means we need to check if ~ is (1) Reflexive (2)Symmetric (3) TransitiveWe begin with (1). This means checking to make sure that for all integers æ, we have æ ~ x. Recall the definitionof ~ for this problem and we see that this is equivalent to saying that for all integers x, we have that (x+x) is aninteger multiple of 7.Is this true? If so, enter Y; if not, enter an integer for which this is falseNext, we check (2). This means checking to make sure that for all integers æ, y we have a ~ y # y ~ x. Unwindthe definition of ~ as we have done for (1) and we see thatx ~ y A x + y = 7m for some integer my ~ æ → y + x = 7m for some integer mBased on that, is (2) true? If so, enter Y; if not, enter a pair of integers for which this is false.Finally, we check (3). This means checking to make sure that for all integers æ, y, z, if æ ~ y and y ~ z thenæ ~ z. Is this true? If so, enter Y; if not, give a triple of integers for which this fails.Finally, based on this calculation, is- an equivalence relation on the set of integers? Enter Y or N.

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