Question

Civen \(A=\\{4,5,6\\}, B=\\{3,4,6,7\\},\) and \(C=\\{2,3,6\\},\) verify the distributive law.

Short answer

The distributive law holds true as both sides yield \{4,6\}.

Step-by-step solution

Understand the Distributive Law

The distributive law in set theory states that for any sets A, B, and C, the following equation holds:\[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \] Our task is to verify this law using the given sets: \( A=\{4,5,6\} \), \( B=\{3,4,6,7\} \), and \( C=\{2,3,6\} \).

Find \(B \cup C\)

Compute the union of sets B and C.\[ B \cup C = \{3,4,6,7\} \cup \{2,3,6\} \]This means we take all unique elements from both sets.\[ B \cup C = \{2,3,4,6,7\} \]

Find \(A \cap (B \cup C)\)

Find the intersection of set A with \(B \cup C\).\[ A \cap (B \cup C) = \{4,5,6\} \cap \{2,3,4,6,7\} \]Only keep elements present in both sets.\[ A \cap (B \cup C) = \{4,6\} \]

Find \(A \cap B\) and \(A \cap C\) Separately

First, find \(A \cap B\):\[ A \cap B = \{4,5,6\} \cap \{3,4,6,7\} = \{4,6\} \]Next, find \(A \cap C\):\[ A \cap C = \{4,5,6\} \cap \{2,3,6\} = \{6\} \]

Find \((A \cap B) \cup (A \cap C)\)

Compute the union of the intersections calculated:\[ (A \cap B) \cup (A \cap C) = \{4,6\} \cup \{6\} \]Combine the elements from both sets, keeping only unique elements.\[ (A \cap B) \cup (A \cap C) = \{4,6\} \]

Compare the Results

Both sides of the distributive law (\( A \cap (B \cup C)\) and \((A \cap B) \cup (A \cap C)\)) result in the set \{4,6\}. This confirms the law:\[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) = \{4,6\} \]

Key concepts

Distributive Law

The Distributive Law is a fundamental principle in set theory, which helps us simplify expressions involving sets. It allows us to express one complicated set operation using simpler operations, making computations easier. The law has a particular form involving intersections and unions of sets. To be precise, it states: \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \).

This means that intersecting a set \( A \) with the union of sets \( B \) and \( C \) is the same as taking the union of the intersections of \( A \) with \( B \) and \( A \) with \( C \). This distributive property is similar to the distributive property of multiplication over addition in arithmetic, where \( a(b + c) = ab + ac \). It's a key concept that reflects how certain operations distribute over others.

• Helps simplify set expressions.
• Analogous to arithmetic distributive law.
• Facilitates computations in set theory.

Understanding this property is crucial, especially when dealing with complex set expressions or proving equality between two sets.

Set Operations

Set operations are the backbone of set theory, just like addition and multiplication are for arithmetic. The fundamental set operations include union, intersection, and difference. Each of these operations serves a unique purpose in set manipulation.

To illustrate:

• Union (\( \cup \)): Combines all unique elements from the involved sets.
• Intersection (\( \cap \)): Keeps only the elements present in all sets.
• Difference (\( - \)): Includes elements that are in one set but not the other.

Every operation has a specific symbol and meaning, making it essential to know what each does to solve set theory problems.

The power of these operations emerges when they're leveraged together, such as in the distributive law or when determining relationships between sets. Mastering these operations will make solving complex set problems far more manageable.

Union and Intersection

The union and intersection are two of the most basic yet powerful operations in set theory. Understanding them deeply can enhance your problem-solving skills in set theory significantly.

The union of sets, symbolized by \( \cup \), refers to combining all unique elements from the involved sets into one set. For example, if \( B = \{3, 4, 6, 7\} \) and \( C = \{2, 3, 6\} \), then their union \( B \cup C \) is \( \{2, 3, 4, 6, 7\} \).

On the other hand, the intersection of sets, symbolized by \( \cap \), involves compiling elements common to all the involved sets. Taking set \( A = \{4, 5, 6\} \) and the union found, \( B \cup C = \{2, 3, 4, 6, 7\} \), their intersection \( A \cap (B \cup C) \) would be \( \{4, 6\} \).

These operations not only define the relationships between sets but are also crucial in algebraic manipulations like verifying the distributive law, validating subsets, or determining disjoint sets.

Mathematical Proofs

Mathematical proofs provide the rigor and foundation in mathematics necessary to understand and establish truths. In terms of set theory, they are used to prove the properties and laws such as the distributive law. Using a series of logical arguments and operations, proofs show why a particular statement holds true under specific conditions.

When verifying the distributive law with the given sets \( A, B, \) and \( C \), a step-by-step approach ensures that every part of the law is accounted for and holds true. This involves:

• Setting the expression as per the distributive law.
• Applying union and intersection operations.
• Comparing both sides of the equation.

Proofs may seem intricate and detailed, but they are paramount in eliminating any ambiguity or errors in mathematical statements. Practicing proofs in set theory strengthens logical reasoning and problem-solving skills, which are valuable in various mathematical applications and beyond.