The question asks to prove that the set of integers $\mathbb{Z}$ is a poset (partially ordered set) with respect to the relation $\geq$ (greater than or equal to).

Proof:

To prove that $\mathbb{Z}$ is a poset under $\geq$, we need to show that the relation $\geq$ satisfies the following three properties:

1. Reflexivity
   For all $a \in \mathbb{Z}$, $a \geq a$.

   • This is true because every integer is greater than or equal to itself.
   • Hence, $\geq$ is reflexive.

2. Antisymmetry
   For all $a, b \in \mathbb{Z}$, if $a \geq b$ and $b \geq a$, then $a = b$.

   • If $a \geq b$, it means $a$ is greater than or equal to $b$.
   • If $b \geq a$, it means $b$ is greater than or equal to $a$.
   • The only way both can be true is if $a = b$.
   • Hence, $\geq$ is antisymmetric.

3. Transitivity
   For all $a, b, c \in \mathbb{Z}$, if $a \geq b$ and $b \geq c$, then $a \geq c$.

   • If $a \geq b$, then $a$ is greater than or equal to $b$.
   • If $b \geq c$, then $b$ is greater than or equal to $c$.
   • This implies $a$ is greater than or equal to $c$.
   • Hence, $\geq$ is transitive.

Conclusion:

Since the relation $\geq$ satisfies reflexivity, antisymmetry, and transitivity, it is a partial order. Therefore, $\mathbb{Z}$ is a poset under the relation $\geq$.

Would you like me to explain any part of this proof further or provide additional examples?

Related Questions:

1. What is the difference between a partial order and a total order?
2. Can $\leq$ be used as a partial order for other sets, like real numbers?
3. How do we prove that a given set with a specific relation is not a poset?
4. Are there other relations besides $\geq$ that make $\mathbb{Z}$ a poset?
5. How does a Hasse diagram represent a poset?

Tip:

When working with posets, always start by checking the three key properties (reflexivity, antisymmetry, and transitivity) for the given relation.