Question

Prove Theorem 1 (show that \(x\) is in the left-hand set iff it is in the right- hand set). For example, for (d), $$ \begin{aligned} x \in(A \cup B) \cap C & \Longleftrightarrow[x \in(A \cup B) \text { and } x \in C] \\ & \Longleftrightarrow[(x \in A \text { or } x \in B), \text { and } x \in C] \\\ & \Longleftrightarrow[(x \in A, x \in C) \text { or }(x \in B, x \in C)]. \end{aligned} $$

Short answer

The theorem is proved by establishing the equivalence in both directions.

Step-by-step solution

Understand the Problem

We are asked to prove that an element \(x\) belongs to the set \((A \cup B) \cap C\) if and only if it belongs to the set \([(x \in A, x \in C) \text{ or }(x \in B, x \in C)]\). This involves showing both directions: \(\rightarrow\) and \(\leftarrow\).

Prove \(\rightarrow\) Direction

Assume \(x \in (A \cup B) \cap C\). By the definition of intersection, \(x\) must be in both \(A \cup B\) and \(C\). Thus, \(x \in C\) and \(x \in (A \cup B)\). Since \(x \in (A \cup B)\), it follows that \(x\) is either in \(A\) or in \(B\) (by the definition of union). Thus, we can conclude that \((x \in A \text{ and } x \in C)\) or \((x \in B \text{ and } x \in C)\).

Prove \(\leftarrow\) Direction

Suppose \((x \in A \text{ and } x \in C)\) or \((x \in B \text{ and } x \in C)\). This means that \(x\) satisfies the condition of being in \(C\), and either in \(A\) or in \(B\). If \(x \in A\), then \(x \in (A \cup B)\); similarly, if \(x \in B\), \(x \in (A \cup B)\). Thus, in either scenario, \(x\) must be in both \(A \cup B\) and \(C\), so \(x \in (A \cup B) \cap C\).

Conclude Equivalence

Since we've proven both directions (\(\rightarrow\) and \(\leftarrow\)), we can conclude that \(x \in (A \cup B) \cap C \Longleftrightarrow [(x \in A, x \in C) \text{ or }(x \in B, x \in C)]\). The sets are equivalent.

Key concepts

Mathematical proof

A mathematical proof is a series of logical statements that demonstrate the truth of a given proposition or theorem. In this specific exercise, we aim to prove a statement about the equivalence of two sets using a series of logical steps. The process involves showing that an element belongs to one set if and only if it belongs to another set.

The phrase "if and only if" often appears in proofs, symbolizing a bidirectional agreement between conditions. When we say that a statement is true "if and only if" another statement is true, we need to prove two separate implications.

• The first direction: assuming the first statement, and logically deducing the second.
• The second direction: assuming the second statement, and logically deducing the first.

Successfully proving both directions confirms the logical equivalence between the statements and demonstrates the validity of the theorem in question.

Intersection and union of sets

Set theory is all about understanding the relationships between different groups, called sets. One core idea involves finding what's common between sets, known as intersection, and merging them, known as union. Let's break these down a bit more:

**Intersection**:

• The intersection of sets, denoted as \(A \cap B\), is the set containing all elements that are common to both \(A\) and \(B\).
• For example, if \(A = \{1, 2, 3\}\) and \(B = \{2, 3, 4\}\), then \(A \cap B = \{2, 3\}\).

**Union**:

• The union of sets, denoted as \(A \cup B\), is the set containing all elements that are either in \(A\), in \(B\), or in both.
• For example, the union of \(A\) and \(B\) from the previous example would give \(A \cup B = \{1, 2, 3, 4\}\).

Understanding these operations helps in proving expressions involving various sets. In the exercise, we use these operations to demonstrate how elements interact within combined sets, reinforcing the idea of logical equivalence.

Logical equivalence

Logical equivalence in set theory means that different expressions can represent the same set. It indicates that two conditions are actually two ways to describe the same scenario. In formal logic, logical equivalence is akin to saying two statements are always either both true or both false.

In our problem, we look at two set expressions and aim to show that they describe the same set of elements through logical deductions. The statements "\(x \in (A \cup B) \cap C\)" and "\((x \in A, x \in C) \text{ or }(x \in B, x \in C)\)" are shown to be equivalent. This is achieved by demonstrating that:

• If an element \(x\) belongs to the first set, it must belong to the second, and vice versa.
• This is akin to showing that one condition's truthfulness inherently guarantees the truthfulness of the other, and they share the same truth values in every possible scenario.

When both directions are proven, it confirms that the statements are logically equivalent, thus completing the proof and establishing the theorem's truth.