Which of the following are possets?

  1. (Z, =)
  2. (Z, ≠)
  3. (Z, ⊆)

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a) (Z, = )

  1. When a ∈ z then a = a and thus relation is reflexive (Here, a = a ∀ a ∈ z)
  2. When a = b and b = a  ∀ a, b ∈ z then a = b and thus relation is antisymmetric
  3. When a = b and b = c  ∀ a, b, c ∈ z then a = c and thus relation is transitive.

So, (Z, =) is a poset.

b) (Z, ≠)

  1. When a ∈ z, then a ≠ a thus relation is not reflexive (Here, a ≠ a ∀ a ∈ z)
  2. When a ≠ b and b ≠ a ∀ a, b ∈ z then a ≠ b and thus relation is not antisymmetric.
  3. When a ≠ b and b ≠ c  ∀ a, b, c ∈ z then a ≠ c and thus relation is not transitive.

So, (Z, ≠) is not a poset.

c) (Z, ⊆)

  1. When a ∈ z, then a ⊆ a thus relation is reflexive (Here, a ⊆ a ∀ a ∈ z)
  2. When a ⊆ b and b ⊆ a ∀ a, b ∈ z then a ⊆ b and thus relation is antisymmetric.
  3. When a ⊆ b and b ⊆ c  ∀ a, b, c ∈ z then a ⊆ c and thus relation is transitive.

So, (Z, ≠) is a poset.

d) (Z, )

  1. When a ≥ z, then a ≥ a thus relation is reflexive (Here, a ≥ a ∀ a ∈ z)
  2. When a ≥ b and b ≥ a ∀ a, b ≥ z then a ≥ b and thus relation is antisymmetric.
  3. When a ≥ b and b ≥ c  ∀ a, b, c ≥ z then a ≥ c and thus relation is transitive.

So, (Z, ≥) is a poset.

Similarly, (Z, ≤), (Z, ÷) are also posets.
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