Let's analyze each binary relation one by one:
A. The relation R on Q (rational numbers) defined as x ~ y if xy = 0.
Reflexivity: For any rational number x, we have x * x = x^2, which can be equal to 0 only if x = 0. Therefore, x ~ x is true only for x = 0. So, the relation R is reflexive.
Symmetry: If x ~ y, then xy = 0. This implies that yx = 0, so y ~ x. Therefore, the relation R is symmetric.
Anti-symmetry: If both x ~ y and y ~ x, then xy = 0 and yx = 0, implying that x = y. Therefore, the relation R is anti-symmetric.
Transitivity: If x ~ y and y ~ z, then xy = 0 and yz = 0. Multiplying these equations, we have xyz^2 = 0, which means x ~ z. Therefore, the relation R is transitive.
Partial order: The relation R satisfies reflexivity, anti-symmetry, and transitivity, making it a partial order.
B. The relation "has the same mother" on the set of students.
Reflexivity: Every student has the same mother as themselves. Therefore, the relation "has the same mother" is reflexive.
Symmetry: If student A has the same mother as student B, then it implies that student B has the same mother as student A. Therefore, the relation "has the same mother" is symmetric.
Anti-symmetry: If two students have the same mother, it does not imply that they are the same person. Therefore, the relation "has the same mother" is not anti-symmetric.
Transitivity: If student A has the same mother as student B and student B has the same mother as student C, then it implies that student A has the same mother as student C. Therefore, the relation "has the same mother" is transitive.
Partial order: The relation "has the same mother" satisfies reflexivity, symmetry, and transitivity, but not anti-symmetry. Therefore, it is not a partial order.
C. The relation R on Q (rational numbers) defined as aRb if ab > 0.
Reflexivity: For any non-zero rational number a, we have a * a = a^2, which is always greater than 0. Therefore, aRa is true for all non-zero a. Additionally, for 0, 0 * 0 = 0, which is not greater than 0. Therefore, 0 does not relate to itself under the relation R. So, the relation R is reflexive except for 0.
Symmetry: If aRb, it means that ab > 0. This implies that ba > 0, so bRa. Therefore, the relation R is symmetric.
Anti-symmetry: If both aRb and bRa, it implies that ab > 0 and ba > 0. This means that a and b have the same sign, and since they are non-zero, it implies that a = b. Therefore, the relation R is anti-symmetric.
Transitivity: If aRb and bRc, it means that ab > 0 and bc > 0. Multiplying these inequalities, we have abc^2 > 0, which means aRc. Therefore, the relation R is transitive.
Partial order: The relation R satisfies reflexivity, anti-symmetry, and transitivity, making it a partial order.
D. The relation of division on the integers.
Reflexivity: Every integer divides itself without leaving a remainder. Therefore, the relation of division is reflexive.
Symmetry: If a divides b, it means that b is divisible by a. This implies that a is also divisible by b, so the relation of division is symmetric.
Anti-symmetry: If a divides b and b divides a, it implies that both a and b are multiples of each other. In this case, it means that a = b or a = -b. Therefore, the relation of division is anti-symmetric if and only if a = b or a = -b.
Transitivity: If a divides b and b divides c, it implies that a is a factor of b and b is a factor of c. Therefore, a must also be a factor of c, meaning that a divides c. Therefore, the relation of division is transitive.
Partial order: The relation of division satisfies reflexivity, anti-symmetry, and transitivity, making it a partial order.
E. The relation {(a, b), (b, c), (a, c)} on the set {a, b, c}.
Reflexivity: For any element x in the set {a, b, c}, (x, x) is present in the relation. Therefore, the relation is reflexive.
Symmetry: If (a, b) is in the relation, it implies that (b, a) is also present. Similarly, if (b, c) is in the relation, then (c, b) is present. However, the presence of (a, c) does not guarantee the presence of (c, a). Therefore, the relation is not symmetric.
Anti-symmetry: If (a, b) and (b, a) are both present in the relation, it implies that a = b. Similarly, if (b, c) and (c, b) are both present, it implies that b = c. Therefore, the relation is anti-symmetric.
Transitivity: If (a, b) and (b, c) are both present in the relation, it implies that (a, c) is also present. Therefore, the relation is transitive.
Partial order: The relation satisfies reflexivity, anti-symmetry, and transitivity, making it a partial order.
F. The relation {(x, x), (y, y)} on (x, y).
Reflexivity: Every element in (x, y) is related to itself by the pairs (x, x) and (y, y). Therefore, the relation is reflexive.
Symmetry: The relation {(x, x), (y, y)} does not contain any pairs where the elements are different. Therefore, it is vacuously symmetric.
Anti-symmetry: The relation {(x, x), (y, y)} contains pairs where the elements are different (x ≠ y), but it is still anti-symmetric because there are no pairs of the form (a, b) and (b, a) for distinct a and b.
Transitivity: The relation {(x, x), (y, y)} contains only pairs where the elements are the same, and transitivity does not apply to those pairs. Therefore, it is transitive.
Partial order: The relation {(x, x), (y, y)} satisfies reflexivity, anti-symmetry, and transitivity, making it a partial order.
To summarize:
A. Reflexive, symmetric, anti-symmetric, transitive, a partial order. B. Reflexive, symmetric, not anti-symmetric, transitive, not a partial order. C. Reflexive (except for 0), symmetric, anti-symmetric, transitive, a partial order. D. Reflexive, symmetric
