Question

Verify that \(\mathrm{P} \times(\mathrm{Q} \cup \mathrm{R})=(\mathrm{P} \times \mathrm{Q}) \cup(\mathrm{P} \times \mathrm{R})\)

Short answer

We have shown that for any element \((p, x)\), if \((p, x) \in P \times(Q \cup R)\) then \((p, x) \in (P \times Q) \cup (P \times R)\) and vice versa. Therefore, we can conclude that \(P \times(Q \cup R) = (P \times Q) \cup (P \times R)\).

Step-by-step solution

Proving \(P \times(Q \cup R) \subseteq (P \times Q) \cup (P \times R)\)

Let's take an element \((p, x) \in P \times(Q \cup R)\). By the definition of the cartesian product, we know that \(p \in P\) and \(x \in (Q \cup R)\). Since \(x \in (Q \cup R)\), it must be the case that \(x \in Q\) or \(x \in R\) (or both). If \(x \in Q\), then \((p, x) \in P \times Q\) and consequently, \((p, x) \in (P \times Q) \cup (P \times R)\) by the definition of the union of sets. Similarly, if \(x \in R\), then \((p, x) \in P \times R\) and \((p, x) \in (P \times Q) \cup (P \times R)\) too. Since \((p, x)\) was an arbitrary element of \(P \times(Q \cup R)\), we have shown that \(P \times(Q \cup R) \subseteq (P \times Q) \cup (P \times R)\).

Proving \((P \times Q) \cup (P \times R) \subseteq P \times(Q \cup R)\)

Now, let's take an element \((p, x) \in (P \times Q) \cup (P \times R)\). By definition of the union of sets, we know that \((p, x) \in P \times Q\) or \((p, x) \in P \times R\) (or both). If \((p, x) \in P \times Q\), then by definition of the cartesian product, \(p \in P\) and \(x \in Q\), so \(x \in (Q \cup R)\). Consequently, \((p, x) \in P \times(Q \cup R)\) by the definition of the cartesian product. Similarly, if \((p, x) \in P \times R\), then by definition of the cartesian product, \(p \in P\) and \(x \in R\), so \(x \in (Q \cup R)\). As a result, \((p, x) \in P \times(Q \cup R)\) too. Since \((p, x)\) was an arbitrary element of \((P \times Q) \cup (P \times R)\), we have shown that \((P \times Q) \cup (P \times R) \subseteq P \times(Q \cup R)\). Since we have proven both directions, we can conclude that: \[P \times(Q \cup R) = (P \times Q) \cup (P \times R)\]

Key concepts

Cartesian Product

The Cartesian Product is a key concept in set theory that combines elements from two sets to form ordered pairs. Imagine you have two sets, \(A\) and \(B\). The Cartesian product, denoted as \(A \times B\), includes every possible ordered pair \((a, b)\) where \(a\) is an element of \(A\) and \(b\) is an element of \(B\). This creates a set of combinations, often visualized in a grid or a table.

To think of it in real-world terms, if \(A\) is a set of shirts and \(B\) is a set of pants, then \(A \times B\) would represent all possible outfit combinations.

The Cartesian Product is essential for understanding more advanced mathematical structures like functions, relations, and databases, making it a fundamental building block in mathematics.

Union of Sets

The Union of Sets operation is one of the basic operations applied to sets, symbolized by \(\cup\). This operation takes two or more sets and combines all their elements into a single set. The result is a set containing every distinct element from the original sets.

For instance, if \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\), the union \(A \cup B\) would result in \(\{1, 2, 3, 4, 5\}\). The key point is that each unique element is included only once in the union, even if it appears in multiple original sets.

This concept helps in various applications such as probability theory, where combining different outcomes is common, or in database queries when merging data from different sources.

Subset

A subset is a concept that deals with comparisons between sets. A set \(A\) is considered a subset of a set \(B\), denoted as \(A \subseteq B\), if every element of \(A\) is also an element of \(B\). This means that \(A\) can either be entirely contained within \(B\) or be equal to \(B\).

For example, if \(B = \{1, 2, 3, 4\}\) and \(A = \{2, 3\}\), then \(A\) is a subset of \(B\) because both elements of \(A\) are found in \(B\).

Subsets are important for defining hierarchical relationships between sets, such as understanding different levels of classification systems or organizing data into more manageable parts.

Proof in Mathematics

Proof in mathematics is a structured way of demonstrating that a certain statement or theorem is true. Unlike empirical evidence in sciences where observations can suggest patterns, mathematical proof provides definitive verifications.

A proof typically follows logical reasoning and starts from accepted truths such as definitions, axioms, or already proven theorems. It then proceeds through step-by-step logical deductions to arrive at the conclusion. The example given here involves proving that \(P \times(Q \cup R) = (P \times Q) \cup (P \times R)\) by demonstrating both subset inclusions.

Mathematical proofs are crucial because they ensure that results are universally valid and grounded in consistent logic, which is essential for the progress and reliability of mathematical knowledge.