Question

Let \(A, B,\) and \(C\) be sets. (a) Prove that if \(A \subset B\) and \(B \subseteq C\), then \(A \subset C\). (b) Prove that if \(A \subseteq B\) and \(B \subset C,\) then \(A \subset C\). (c) Prove that if \(A \subseteq B\) and \(A \not \subseteq C,\) then \(B \nsubseteq C\).

Short answer

(a) and (b) follow from the transitive property of subset relations; (c) follows by contradiction.

Step-by-step solution

Understanding the Subset Relation

In a subset relationship, a set \(X\) is a subset of \(Y\) (denoted \(X \subseteq Y\)) if every element of \(X\) is also an element of \(Y\). If \(X \subset Y\), then \(X \subseteq Y\) and there is at least one element in \(Y\) that is not in \(X\).

Prove (a)

Assume \(A \subset B\), meaning every element of \(A\) is in \(B\), and \(B \subseteq C\), meaning every element of \(B\) is in \(C\). Then, every element of \(A\) is also in \(C\), thus \(A \subset C\). This follows directly by the transitive property of the subset relation.

Prove (b)

Assume \(A \subseteq B\), meaning every element of \(A\) is in \(B\), and \(B \subset C\), meaning every element of \(B\) is in \(C\) but \(B\) is not equal to \(C\). Therefore, every element of \(A\) is also in \(C\), so \(A \subset C\). The fact that \(B\) is a proper subset of \(C\) does not affect the elements of \(A\).

Prove (c)

Assume \(A \subseteq B\) and \(A ot\subseteq C\), meaning there exists at least one element in \(A\) that is not in \(C\). Since \(A \subseteq B\), this element must also be in \(B\). Therefore, it shows that not every element of \(B\) is in \(C\), implying \(B subseteq C\).

Key concepts

Subset Relation

In set theory, the concept of subset relations is pivotal in understanding how sets interact with each other. A set \(X\) is said to be a subset of set \(Y\), denoted as \(X \subseteq Y\), if every element found in \(X\) is also found within \(Y\). This relationship implies that \(Y\) encompasses all elements of \(X\), but may contain additional elements not present in \(X\). There's also an idea of a proper subset, where \(X \subset Y\) signifies that \(X\) is not only a subset but is strictly contained within \(Y\), meaning there exists at least one element in \(Y\) that's not in \(X\).

• \(X \subseteq Y\): All elements of \(X\) are in \(Y\).
• \(X \subset Y\): \(X\) is a proper subset of \(Y\), indicating \(X\) is not equivalent to \(Y\).

Being able to identify and work with these kinds of subset relations is essential for solving more complex problems in set theory.

Transitive Property

The transitive property is a fundamental principle in mathematics and set theory. It states that if two relationships are true, a third related relationship emerges from them. Regarding subsets, if we know that \(A \subset B\) and \(B \subseteq C\), then it follows that \(A \subset C\). Let's break it down:- First, \(A \subset B\) indicates every element of set \(A\) is in set \(B\).- Second, \(B \subseteq C\) tells us every element of \(B\) is also in \(C\).- Therefore, by combining these facts, any element in \(A\) is also found in \(C\), thus \(A \subset C\).This transitive nature of subsets allows us to simplify complex relationships by understanding how they align through intermediary sets.

Element Inclusion

Element inclusion in set theory refers to the membership of elements within a particular set. This concept is paramount when considering whether one set is a subset of another. If an element is included in one set and that set is included in another, it implies that the element exists in all sets involved in the subset chain.Consider the step where \(A \subseteq B\) and \(A ot \subseteq C\). Since there must be an element of \(A\) not in \(C\), but \(A \subseteq B\), this element also belongs to \(B\). This internal membership or inclusion establishes a connection allowing us to consider relative membership across multiple sets.

• \(A \subseteq B\): The inclusion of elements of \(A\) in \(B\).
• \(A ot \subseteq C\): At least one element of \(A\) is not in \(C\).

Understanding element inclusion helps clarify the internal workings of subset relationships and their logical proofs.

Proper Subset

A proper subset is a subset that is strictly contained within another set and is not identical to the larger set. Denoted \(A \subset B\), it implies all elements of \(A\) are in \(B\) but there exists at least one element in \(B\) that is not in \(A\). This distinction is important in set theory as it explains hierarchy within sets. For example, \(B \subset C\) highlights that while \(B\) is entirely contained within \(C\), there are additional elements in \(C\) that make it larger.

• \(A \subset B\): All elements in \(A\) exist in \(B\) and \(A\) is strictly smaller.
• Explanation: \(B\) consists of every element \(A\) has, plus more.

Recognizing proper subsets allows for more nuanced discussions about the containment and equivalence of different sets.