Problem 5 Which of these are posets? a) ... [FREE SOLUTION]

Chapter 9: Problem 5

Which of these are posets? a) \((\mathbf{Z},=)\) b) \((\mathbf{Z}, \neq)\) c) \((\mathbf{Z}, \geq)\) d) \((\mathbf{Z}, \not{|})\)

Short Answer

Expert verified

Relations (a) and (c) are posets.

Step by step solution

Understand the definition of a poset

A poset, or partially ordered set, is a set equipped with a binary relation that is reflexive, antisymmetric, and transitive.

Check reflexivity of each relation

A relation is reflexive if for all elements a in the set, the relation aRa holds.

Step 2a: Check reflexivity of (\(\mathbf{Z}, =\))

The equality relation (\(=\)) in \(\mathbf{Z}\) is reflexive because every integer is equal to itself. Therefore, this relation is reflexive.

Step 2b: Check reflexivity of (\(\mathbf{Z}, eq\))

The inequality relation (\(eq\)) is not reflexive, because it is not true that any integer is not equal to itself. Therefore, this relation is not reflexive.

Step 2c: Check reflexivity of (\(\mathbf{Z}, \geq\))

The relation \(\geq\) in \(\mathbf{Z}\) is reflexive because every integer is greater than or equal to itself. Therefore, this relation is reflexive.

Step 2d: Check reflexivity of (\(\mathbf{Z}, ot{|}\))

The relation 'not divides' (\(ot{|}\)) is not reflexive, because it is not true that any integer does not divide itself. Therefore, this relation is not reflexive.

Check antisymmetry of each relation

A relation is antisymmetric if for all a and b in the set, if aRb and bRa, then a = b.

Step 3a: Check antisymmetry of (\(\mathbf{Z}, =\))

The equality relation (\(=\)) is antisymmetric because if a = b and b = a, then a = b trivially holds.

Step 3b: Check antisymmetry of (\(\mathbf{Z}, eq\))

The inequality relation (\(eq\)) cannot be antisymmetric because if a \(eq\) b and b \(eq\) a, then a \(eq\) b is not equivalent to a = b.

Step 3c: Check antisymmetry of (\(\mathbf{Z}, \geq\))

The relation \(\geq\) is antisymmetric because if a \(\geq\) b and b \(\geq\) a, then a = b must hold.

Step 3d: Check antisymmetry of (\(\mathbf{Z}, ot{|}\))

The relation 'not divides' (\(ot{|}\)) is antisymmetric if it is not the case that both a does not divide b and b does not divide a unless a = b.

Check transitivity of each relation

A relation is transitive if for all a, b, and c in the set, if aRb and bRc, then aRc holds.

Step 4a: Check transitivity of (\(\mathbf{Z}, =\))

Equality (\(=\)) is transitive because if a = b and b = c, then we have a = c.

Step 4b: Check transitivity of (\(\mathbf{Z}, eq\))

The inequality relation (\(eq\)) is not transitive because if a \(eq\) b and b \(eq\) c, it does not necessarily follow that a \(eq\) c.

Step 4c: Check transitivity of (\(\mathbf{Z}, \geq\))

The relation \(\geq\) is transitive because if a \(\geq\) b and b \(\geq\) c, then a \(\geq\) c.

Step 4d: Check transitivity of (\(\mathbf{Z}, ot{|}\))

The relation 'not divides' (\(ot{|}\)) is not transitive. For example, 3 does not divide 6, and 6 does not divide 12, but 3 divides 12.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

poset

A poset, short for 'partially ordered set', is a fundamental concept in mathematics. It consists of a set along with a binary relation that satisfies three important properties: reflexivity, antisymmetry, and transitivity. This type of structure helps us understand and work with ordered relationships within a set.

To break it down:

Reflexivity means every element is related to itself.
Antisymmetry means if one element is related to another, and vice versa, they must be identical.
Transitivity means if one element is related to a second, and the second to a third, then the first is also related to the third.

Let's dive deeper into these properties to understand how they define a poset.

reflexivity

Reflexivity is the property of a relation where each element is related to itself. In other words, for a relation \( R \) and a set \( S \), \( R \) is reflexive if for every element \( a \) in \( S \), the relation \( aRa \) holds.

Let's examine reflexivity in the context of our examples:

For \((\mathbf{Z}, =)\), the relation is reflexive because any integer is equal to itself.
For \((\mathbf{Z}, eq)\), the relation is not reflexive since an integer cannot be unequal to itself.
For \((\mathbf{Z}, \geq)\), the relation is reflexive because any integer is greater than or equal to itself.
For \((\mathbf{Z}, ot{|})\), 'not divides' is not reflexive since it is not true that any integer does not divide itself.

Grasping reflexivity helps in understanding the fundamental framework that helps build a poset.

antisymmetry

Antisymmetry is a crucial property in a poset where if one element is related to another, and vice versa, then they must be the same element. Formally, a relation \( R \) on a set \( S \) is antisymmetric if for all \( a, b \) in \( S \), whenever \( aRb \) and \( bRa \) hold, it must follow that \( a = b \).

Let鈥檚 review antisymmetry in our examples:

For \((\mathbf{Z}, =)\), the relation is trivially antisymmetric because if \( a = b \) and \( b = a \), then obviously \( a = b \).
For \((\mathbf{Z}, eq)\), the relation cannot be antisymmetric because if \( a eq b \) and \( b eq a \), \( a eq b \) does not imply \( a = b \).
For \((\mathbf{Z}, \geq)\), the relation is antisymmetric because if \( a \geq b \) and \( b \geq a \), then \( a = b \) must hold.
For \((\mathbf{Z}, ot{|})\), the relation 'not divides' is antisymmetric because if both \( a ot{|} b \) and \( b ot{|} a \), then typically \( a = b \).

Understanding antisymmetry helps detect the orderly nature within the structure of a poset.

transitivity

Transitivity is another key property of a poset, stating that if one element is related to a second, and the second element is related to a third, then the first must also be related to the third. Formally, a relation \( R \) on a set \( S \) is transitive if for every \( a, b, \) and \( c \) in \( S \), whenever \( aRb \) and \( bRc \), then \( aRc \) must hold.

Here's how transitivity works in our cases:

For \((\mathbf{Z}, =)\), it鈥檚 transitive because if \( a = b \) and \( b = c \), then we indeed have \( a = c \).
For \((\mathbf{Z}, eq)\), it鈥檚 not transitive because if \( a eq b \) and \( b eq c \), it doesn't necessarily follow that \( a eq c \).
For \((\mathbf{Z}, \geq)\), it鈥檚 transitive because if \( a \geq b \) and \( b \geq c \), then \( a \geq c \).
For \((\mathbf{Z}, ot{|})\), 'not divides' is not transitive. For instance, 3 does not divide 6, and 6 does not divide 12, but 3 divides 12.

Grasping transitivity rounds up the understanding of key properties needed for establishing posets.

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