Question

Let \((a, b)\) denote the set $$ \\{\\{a\\},\\{a, b\\}\\} $$ (Kuratowski's definition of an ordered pair). (i) Which of the following statements are true? (a) \(a \in(a, b)\); (b) \(\\{a\\} \in(a, b)\); (c) \((a, a)=\\{a\\}\); (d) \(b \in(a, b)\); (e) \(\\{b\\} \in(a, b)\); \((\mathrm{f})\\{a, b\\} \in(a, b)\); (ii) Prove that \((a, b)=(u, v)\) iff \(a=u\) and \(b=v\). [Hint: Consider separately the two cases \(a=b\) and \(a \neq b,\) noting that \(\\{a, a\\}=\) \(\\{a\\} .\) Also note that \(\\{a\\} \neq a .]\)

Short answer

Statements (b) and (f) are true; others are false. An ordered pair is equal if and only if the corresponding elements are equal.

Step-by-step solution

Understanding Kuratowski's Definition

According to Kuratowski's definition, an ordered pair \((a, b)\) is defined as the set \(\{\{a\},\{a, b\}\}\). This means it contains two elements: the singleton set \(\{a\}\) and the pair set \(\{a, b\}\).

Analyzing Statement (a)

Statement (a) asks if \(a \in (a, b)\). Based on the definition, the set \((a, b) = \{\{a\}, \{a, b\}\}\), contains \{a\} as an element, but not \(a\) by itself. Thus, \(a otin (a, b)\).

Analyzing Statement (b)

Statement (b) asks if \(\{a\} \in (a, b)\). The set \((a, b)\) explicitly includes \(\{a\}\) as one of its elements, so \(\{a\} \in (a, b)\).

Analyzing Statement (c)

Statement (c) posits that \((a, a) = \{a\}\). According to the definition, \((a, a) = \{\{a\}, \{a, a\}\}\), which simplifies to \(\{\{a\}\}\), not \(\{a\}\). Therefore, the statement is false.

Analyzing Statement (d)

Statement (d) asks if \(b \in (a, b)\). Since neither \(\{a\}\) nor \(\{a, b\}\) contains \(b\) itself but only as part of \{a, b\}, \(b otin (a, b)\).

Analyzing Statement (e)

Statement (e) asks if \(\{b\} \in (a, b)\). \(\{b\}\) does not appear in the pair, so \(\{b\} otin (a, b)\).

Analyzing Statement (f)

Statement (f) asks if \(\{a, b\} \in (a, b)\). The set \{a, b\} is explicitly one of the two elements of \((a, b)\), so \(\{a, b\} \in (a, b)\).

Proving Ordered Pair Equality (Case a = b)

If \(a = b\), then \((a, b) = \{\{a\}\}\). Similarly, \((u, v) = \{\{u\}\}\) if \(u = v\). These are equal when \(a = u\) and \(b = v\).

Proving Ordered Pair Equality (Case a \neq b)

For \(a eq b\), \((a, b) = \{\{a\}, \{a, b\}\}\). If \((a, b) = (u, v)\), then \{a\}\ is \{u\}\ and \{a, b\}\ is \{u, v\}\, yielding \(a = u\) and \(b = v\).

Final Equality Proof

In both cases \((a, b) = (u, v)\) if and only if \(a = u\) and \(b = v\), concluding the proof of ordered pair equality.

Key concepts

Kuratowski's Definition

Kuratowski's definition of an ordered pair provides a way to define ordered pairs using set theory concepts. According to this definition, the ordered pair \((a, b)\) is represented as the set \(\{\{a\}, \{a, b\}\}\). This is a simple yet powerful way to encapsulate the idea of an ordered pair within the framework of sets.

This definition ensures that two ordered pairs \((a, b)\) and \((u, v)\) are equal if and only if \(a = u\) and \(b = v\). It achieves this by the specific arrangement of elements within the set.

Let's break it down:
- The first part, \(\{a\}\), ensures that the first element in the pair is represented uniquely.
- The second part, \(\{a, b\}\), captures the relationship between the first and second elements.

By using sets in this particular way, Kuratowski's definition avoids ambiguity and clearly distinguishes between \((a, b)\) and other possible arrangements using the same elements.

Set Theory

Set theory is the mathematical study of sets, which are collections of objects. Set theory provides the foundational language and framework used throughout mathematics. It's fundamentally important because it allows us to discuss collections of elements and their relationships in a structured way.

Kuratowski's definition of ordered pairs is a direct application of set theory. By defining an ordered pair as a set, it harnesses the properties of sets to address questions of order and uniqueness in pairs.

Some important concepts in set theory that relate to Kuratowski's definition include:
- **Elements and sets:** In set theory, we talk about objects that are either elements or sets themselves. For instance, in \((a, b) = \{\{a\}, \{a, b\}\}\), both \(\{a\}\) and \(\{a, b\}\) are sets.
- **Membership:** We use the symbol \( \in \) to denote membership. For example, \(a \in \{a, b\}\) signifies that \(a\) is a part of the set \(\{a, b\}\).
- **Equality of sets:** Two sets are equal if they contain the exact same elements. This means set \(A\) is equal to set \(B\) if every element of \(A\) is in \(B\) and vice versa.

Mathematical Proofs

Mathematical proofs are logical arguments that verify the truth of a mathematical statement. They involve a sequence of statements, each justified by axioms, definitions, or previously proven results to conclude that a certain statement is true.

In the context of Kuratowski's definition, proving that two ordered pairs \((a, b)\) and \((u, v)\) are equal involves checking that they satisfy the condition: \((a, b) = (u, v)\) if and only if \(a = u\) and \(b = v\). This involves analyzing different cases, such as when the elements of the pair are equal or distinct.

Practical steps in this proof include:
- **Case 1: When \(a = b\):** Here, the ordered pair \((a, b)\) simplifies to \(\{\{a\}\}\). If \((u, v) = \{\{u\}\}\), then \(a = u\) and \(b = v\) ensure equality.
- **Case 2: When \(a eq b\):** We examine the two conditions \(\{a\} = \{u\}\) and \(\{a, b\} = \{u, v\}\) which bring us back to our equality conditions \(a = u\) and \(b = v\).

Through logical reasoning and application of definitions, proofs help us establish the veracity of these mathematical properties, making them a crucial part of advanced mathematics.