Let's analyze each binary relation one by one:
A. A(x, y): true if and only if y is even.
Reflexivity: A relation is reflexive if every element is related to itself. In this case, for any natural number x, A(x, x) is true if and only if x is even. Since every natural number is either even or odd, A(x, x) is true for all x. Therefore, the relation A is reflexive.
Symmetry: A relation is symmetric if whenever A(x, y) is true, A(y, x) is also true. In this case, if y is even, then x must also be even for A(x, y) to be true. Similarly, if x is even, then y must also be even for A(y, x) to be true. Thus, A(x, y) is symmetric.
Anti-symmetry: A relation is anti-symmetric if whenever A(x, y) and A(y, x) are both true, it implies that x = y. In this case, if both x and y are even, then A(x, y) and A(y, x) are both true. However, this does not imply that x = y, as there are multiple even numbers. For example, A(2, 4) and A(4, 2) are both true, but 2 is not equal to 4. Therefore, the relation A is not anti-symmetric.
Transitivity: A relation is transitive if whenever A(x, y) and A(y, z) are both true, it implies that A(x, z) is also true. In this case, if y is even and z is even, then A(y, z) is true. If x is even and y is even, then A(x, y) is true. From these premises, we can conclude that x is even, and thus A(x, z) is also true. Therefore, the relation A is transitive.
Partial order: A partial order is a relation that is reflexive, anti-symmetric, and transitive. Since the relation A is reflexive and transitive but not anti-symmetric, it is not a partial order.
B. B(x, y): true if and only if x < y.
Reflexivity: For any natural number x, B(x, x) is false because x is not less than itself. Therefore, the relation B is not reflexive.
Symmetry: If x < y, then it is not true that y < x unless x = y. Therefore, the relation B is not symmetric.
Anti-symmetry: If both x < y and y < x are true, then it implies that x = y. Therefore, the relation B is anti-symmetric.
Transitivity: If x < y and y < z, then it implies that x < z. Therefore, the relation B is transitive.
Partial order: The relation B satisfies reflexivity, anti-symmetry, and transitivity, making it a partial order.
C. C(x, y): true if and only if x divides y.
Reflexivity: For any natural number x, C(x, x) is true because x divides itself. Therefore, the relation C is reflexive.
Symmetry: If x divides y, then it is not necessarily true that y divides x unless x = y. Therefore, the relation C is not symmetric.
Anti-symmetry: If both x divides y and y divides x are true, then it implies that x = y. Therefore, the relation C is anti-symmetric.
Transitivity: If x divides y and y divides z, then it implies that x divides z. Therefore, the relation C is transitive.
Partial order: The relation C satisfies reflexivity, anti-symmetry, and transitivity, making it a partial order.
D. D(x, y): true if and only if x < y.
Reflexivity: For any natural number x, D(x, x) is false because x is not less than itself. Therefore, the relation D is not reflexive.
Symmetry: If x < y, then it is not true that y < x unless x = y. Therefore, the relation D is not symmetric.
Anti-symmetry: If both x < y and y < x are true, then it implies that x = y. Therefore, the relation D is anti-symmetric.
Transitivity: If x < y and y < z, then it implies that x < z. Therefore, the relation D is transitive.
Partial order: The relation D satisfies anti-symmetry and transitivity but not reflexivity, so it is not a partial order.
E. E(x, y): true if and only if the English name of x comes no later than the name of y in alphabetical order.
Reflexivity: For any natural number x, E(x, x) is true because the English name of x is the same as itself. Therefore, the relation E is reflexive.
Symmetry: If the English name of x comes no later than the name of y, it does not necessarily mean that the name of y comes no later than the name of x unless x = y. Therefore, the relation E is not symmetric.
Anti-symmetry: If both the English name of x comes no later than the name of y and the name of y comes no later than the name of x, then it implies that x = y. Therefore, the relation E is anti-symmetric.
Transitivity: If the English name of x comes no later than the name of y and the name of y comes no later than the name of z, then it implies that the English name of x comes no later than the name of z. Therefore, the relation E is transitive.
Partial order: The relation E satisfies reflexivity, anti-symmetry, and transitivity, making it a partial order.
To summarize:
A. Reflexive, symmetric, not anti-symmetric, transitive, not a partial order.
B. Not reflexive, not symmetric, anti-symmetric, transitive, a partial order.
C. Reflexive, not symmetric, anti-symmetric, transitive, a partial order.
D. Not reflexive, not symmetric, anti-symmetric, transitive, not a partial order.
E. Reflexive, not symmetric, anti-symmetric, transitive, a partial order.
