Explanation:
Let's analyze each of the given relations R in terms of being reflexive, anti-reflexive, symmetric, anti-symmetric, and transitive:
(a) R=A={(x,y)∈Z×Z ∣ x+2y=1/3}:
- Reflexive: This relation is not reflexive because, for any integer x, the pair (x, x) does not satisfy the equation x + 2x = 1/3.
- Anti-reflexive: It is not anti-reflexive because there are pairs (x, x) that do not satisfy the given condition.
- Symmetric: This relation is symmetric because if (x, y) is in R, then (y, x) is also in R, as changing the order of x and y doesn't affect whether they satisfy the equation.
- Anti-symmetric: It is not anti-symmetric because there exist pairs (x, y) and (y, x) in R where x ≠ y.
- Transitive: This relation is not transitive because having (x, y) and (y, z) in R does not necessarily imply (x, z) is in R.
(b) R=B=(x,y)∈Z×Z ∣ :
- Reflexive: This relation is reflexive because for any integer x, the pair (x, x) satisfies the equation |x| + |x| = 4.
- Anti-reflexive: It is not anti-reflexive because it satisfies the reflexive property.
- Symmetric: This relation is symmetric because if (x, y) is in R, then (y, x) is also in R.
- Anti-symmetric: It is not anti-symmetric because there exist pairs (x, y) and (y, x) in R where x ≠ y.
- Transitive: This relation is not transitive because having (x, y) and (y, z) in R does not necessarily imply (x, z) is in R.
(c) R=C={(x,y)∈R×R ∣ xy≤1}:
- Reflexive: This relation is reflexive because for any real number x, the pair (x, x) satisfies the condition xy ≤ 1.
- Anti-reflexive: It is not anti-reflexive because it satisfies the reflexive property.
- Symmetric: This relation is symmetric because if (x, y) is in R, then (y, x) is also in R.
- Anti-symmetric: It is not anti-symmetric because there exist pairs (x, y) and (y, x) in R where x ≠ y.
- Transitive: This relation is transitive because if (x, y) and (y, z) are in R, then (x, z) is also in R due to the multiplication property of real numbers.
(d) R=D=>1:
- Reflexive: This relation is not reflexive because, for any rational number x, the pair (x, x) does not satisfy the condition |x + x| > 1.
- Anti-reflexive: This relation is anti-reflexive because it does not satisfy the reflexive property.
- Symmetric: This relation is not symmetric because there may be pairs (x, y) in R where (y, x) is not in R.
- Anti-symmetric: It is anti-symmetric because if (x, y) and (y, x) are in R, then x + y > 1 and y + x > 1, which implies x = y. Therefore, (x, y) = (y, x) only when x = y.
- Transitive: This relation is not transitive because having (x, y) and (y, z) in R does not necessarily imply (x, z) is in R.