5. Define a relation R on the set of real numbers by (x, y) ER if and only if |1 – y <3. Is this relation reflexive? Is it symmetric? Is it transitive? Is this an equivalence relation? Justify your arguments.

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6. Let be a relation on Z defined as follows: a b if and only if 2a+36 is divisible by 5. (a) Show that is an equivalence relation. (b) What is the equivalence class of 0 ?

why is "x − y is a rational number" is not antisymmetric while "x = 2y" is antisymmetric

the proof I found for the first one is that "Since (1, −1) and (−1, 1) are both in R,
the relation is not antisymmetric." While the proof for the other was, "To see that it is antisymmetric, suppose that x = 2y, and y = 2x. Then y = 4y, from
which it follows that y = 0 and hence x = 0. Thus the only time that (x, y) and (y, x) are both in R
is when x = y (and both are 0)".
now the question is, why is it enough for the first equation to give an example that lies outside the relation to proving that it is NOT antisymmetric. Whereas for the second equation it was enough to give one example to prove that it is antisymmetric, even though in the relation R there exists element which is not antisymmetric

chocolaterat864

[5] 8. Let V = {x,y} be a set with exactly two vectors, r and y. Define vector addition and scalar multiplication in V by the following rules: Vector addition: 2+r= x, y + y = 1, + y = y, and y + x = y. Scalar multiplication: cx = I, and cy = y for all c ER Prove that V is not a vector space by finding one axiom in the definition of a vector space that fails to hold. You must state the axiom clearly and show it does not hold.

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5. Define a relation R on the set of real numbers by (x, y) ER if and only if |1 – y <3. Is this relation reflexive? Is it symmetric? Is it transitive? Is this an equivalence relation? Justify your arguments.

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