- (Z, =)
- (Z, ≠)
- (Z, ⊆)

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Login Now**a) (Z, = )**

- When a ∈ z then a = a and thus relation is reflexive (Here, a = a ∀ a ∈ z)
- When a = b and b = a ∀ a, b ∈ z then a = b and thus relation is antisymmetric
- 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, ≠)**

- When a ∈ z, then a ≠ a thus relation is not reflexive (Here, a ≠ a ∀ a ∈ z)
- When a ≠ b and b ≠ a ∀ a, b ∈ z then a ≠ b and thus relation is not antisymmetric.
- 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, ⊆)**

- When a ∈ z, then a ⊆ a thus relation is reflexive (Here, a ⊆ a ∀ a ∈ z)
- When a ⊆ b and b ⊆ a ∀ a, b ∈ z then a ⊆ b and thus relation is antisymmetric.
- 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, ≥)**

- When a ≥ z, then a ≥ a thus relation is reflexive (Here, a ≥ a ∀ a ∈ z)
- When a ≥ b and b ≥ a ∀ a, b ≥ z then a ≥ b and thus relation is antisymmetric.
- 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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