For each relation, indicate whether the relation is:

• reflexive, anti-reflexive, or neither
• symmetric, anti-symmetric, or neither
• transitive or not transitive

(a) The domain of the relation L is the set of all real numbers. Forx, y R,xLy ifx < y.

(b) The domain of the relation E is the set of all real numbers. Forx, y R,xEy ifx y.

(c) The domain of relation P is the set of all positive integers. Forx, y Z⁺,xPy if there is a positive integer n such thatxⁿ= y.

(d) The domain for the relation D is the set of all integers. For any two integers, x and y, xDy if x evenly divides y. An integer x evenly divides y if there is another integer n such thaty = xn.(Note that the domain is the set of all integers, not just positive integers.)

(e) The domain for the relation A is the set of all real numbers. xAy if|x - y| 2.

(f) The domain for relation R is the set of all real numbers. xRy ifx - yis rational. A real number r is rational if there are two integers a and b, such thatb 0andr = a/b.You can use the fact that the sum of two rational numbers is also rational.

(g) The domain for the relation isZZ.(a, b)is related to(c, d)ifa candb d.

(h) The domain for the relation isZZ.(a, b)is related to(c, d)ifa corb d(inclusive or).

(i) The domain for relation T is the set of real numbers. xTy ifx + y = 0.

(j) The domain for relation Z is the set of real numbers. xZy ify = 2x.

(k) The domain for relation T is a group of people. xTy if person y is taller than person x. There are at least two people in the group who are not the same height.

(l) The domain for relation C is a group of people. xCy if person x is the first cousin of person y (i.e., a parent of person x is a sibling of a parent of person y). You can assume that there at least two people x and y such that x is the first cousin of y. You can also assume that no one has two parents who are siblings of each other.