Question

Show that if \(X \subseteq Y,|X| \leq|Y|\).

Short answer

If \(X \subseteq Y\), then \(|X| \leq |Y|\) based on the definition of subsets.

Step-by-step solution

Understand the Problem Statement

We are given two sets, \(X\) and \(Y\), and it is given that \(X\subseteq Y\). We need to prove that under this condition, the number of elements (cardinality) in set \(X\) is less than or equal to the number of elements in set \(Y\), symbolically: \(|X| \leq |Y|\).

Definition of a Subset

By definition, a set \(X\) is a subset of set \(Y\), denoted as \(X \subseteq Y\), if every element of \(X\) is also an element of \(Y\). This means all elements from \(X\) are contained within \(Y\), and potentially \(X\) has equal or fewer elements compared to \(Y\).

Analyze Set Relations

Since \(X \subseteq Y\), every element of \(X\) 'matches' with an element of \(Y\). Consequently, there cannot be more elements in \(X\) than in \(Y\), thereby ensuring that \(|X|\) is at most \(|Y|\).

Conclude the Proof

Based on the subset definition and the pairing of elements, it is conclusive that \(|X|\) must be less than or equal to \(|Y|\). Thus, our statement \(|X| \leq |Y|\) holds true as required by the assumption \(X \subseteq Y\).

Key concepts

Subsets

In set theory, a *subset* is a fundamental concept. When we say that a set \( X \) is a subset of set \( Y \), written as \( X \subseteq Y \), it means every element in \( X \) is also contained in \( Y \). This relationship implies that \( X \) is either equal to \( Y \) or contains fewer elements.Some important points to remember about subsets:

• If \( X \subseteq Y \), \( X \) may be equal to \( Y \) or have fewer elements.
• The empty set \( \emptyset \) is a subset of every set, including itself.
• If \( X \) is a proper subset of \( Y \), then \( X eq Y \) and \( X \) contains fewer elements.

Understanding subsets helps to visualize how elements of a set can be organized or related to one another.

Cardinality

*Cardinality* describes the number of elements in a set. It is a measure of the "size" of the set. For instance, if set \( X \) has 3 elements, we denote its cardinality by \(|X| = 3\).When considering subsets:

• If \( X \subseteq Y \), then the cardinality of \( X \) is less than or equal to that of \( Y \), i.e., \(|X| \leq |Y|\).
• If \( X \) is a proper subset of \( Y \), then \(|X| < |Y|\).

Cardinality is a key topic when dealing with finite sets in discrete mathematics. It helps in understanding and comparing the sizes of different sets and has wide applications in counting problems.

Discrete Mathematics

*Discrete mathematics* is a branch of mathematics dealing with structures that are distinct and separate, often involving finite or countable sets. It is essential for understanding the foundations of computer science and algorithms.Key areas involving discrete mathematics include:

• Set theory, which forms the basis for many topics in discrete math.
• Combinatorics, which involves counting, arranging, and finding patterns.
• Graph theory, dealing with networks of points and lines.

In discrete mathematics, problems often require logical reasoning and methodical proofs, such as demonstrating that \(|X| \leq |Y|\) when \(X \subseteq Y\). This area of study provides the tools necessary to solve problems involving finite structures logically and efficiently.