Problem 17 If \(A, B\) and \(C\) are sets, ... [FREE SOLUTION]

Chapter 8: Problem 17

If \(A, B\) and \(C\) are sets, then \(A \times(B \cap C)=(A \times B) \cap(A \times C)\)

Short Answer

Expert verified

If \(A\), \(B\) and \(C\) are sets, then it is indeed true that \(A \times (B \cap C) = (A \times B) \cap (A \times C)\). This has been proved by showing that each side of the equation is a subset of the other.

Step by step solution

Understanding The Terms

In set theory, the Cartesian product of \(A\) and \(B\), denoted by \(A \times B\), is the set of all ordered pairs \((a, b)\) where \(a\) is in \(A\) and \(b\) is in \(B\). The intersection of \(B\) and \(C\) is denoted by \(B \cap C\) and represents the set of elements that are in both \(B\) and \(C\). We're asked to prove that \(A \times (B \cap C) = (A \times B) \cap (A \times C)\).

Prove the Left side Subset of Right side

The challenge here is to prove this equality by showing that each side of the equation is a subset of the other. Let us take an arbitary element (a, b) from the left side i.e, \(A \times (B \cap C)\). By definition of the Cartesian product and intersection, \(a \in A\) and \(b \in B \cap C\). Hence, \(b\) is in both \(B\) and \(C\). Therefore, \((a, b) \in A \times B\) and \((a, b) \in A \times C\), so \((a, b) \in (A \times B) \cap (A \times C)\). Therefore, \(A \times (B \cap C)\) is a subset of \((A \times B) \cap (A \times C)\).

Prove the Right side Subset of Left side

Now take an arbitrary element \((a, b)\) from the right side i.e, \((A \times B) \cap (A \times C)\). This implies that \((a, b) \in A \times B\) and \((a, b) \in A \times C\). Therefore, \(a \in A\) and \(b \in B\) and \(b \in C\). This means \(b \in B \cap C\). Hence, \((a, b) \in A \times (B \cap C)\). Therefore, \((A \times B) \cap (A \times C)\) is a subset of \(A \times (B \cap C)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Set theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. These objects are often called elements or members of the set. Understanding how sets interact is crucial for foundational mathematics. Here are some key concepts related to set theory:

Union: The union of two sets includes all elements that are in either set or both.
Intersection: The intersection contains only the elements found in both sets.
Difference: The difference is the set of elements that are in one set but not the other.
Cartesian Product: For sets A and B, the Cartesian product is the set of all ordered pairs \((a, b)\) where \(a\) is in A and \(b\) is in B.

The main focus in set theory is how these operations and definitions interact, which helps in understanding more complex mathematical structures. Regularly, operations like union and intersection help identify relationships between different sets, crucial for proofs and advanced mathematical reasoning.

Intersection of sets

The intersection of sets is an operation that helps us find common elements between two or more sets. It is denoted by the symbol \(\cap\). If we have two sets B and C, their intersection \(B \cap C\) is a new set consisting of elements that exist in both B and C.
Understanding intersections is fundamental in scenarios such as:

Finding common solutions or elements shared between different groups.
Solving problems where we need to focus only on shared characteristics.
Forming the basis for proving complex relationships in set theory.

In practice, the intersection can be visualized as overlapping areas in a Venn diagram, showing the shared components. This simplicity of visualizing intersections helps in grasping more complex logical relationships in mathematics.

Subset

A subset is a set where all its elements are contained within another set. If every element of set A is also in set B, we say that A is a subset of B, denoted as \(A \subseteq B\).
Subsets are essential for understanding how sets relate and interact with each other. They form the building blocks for many mathematical proofs and logical arguments. Here are some attributes of subsets:

A set is always a subset of itself.
The empty set is a subset of every set.
Proper subset, denoted \(A \subset B\), indicates all elements of A are in B, but B contains additional elements not in A.

The concept of subsets is vital in proofs, such as demonstrating equality between sets by proving they are subsets of each other, or in understanding hierarchies and inclusion within mathematical systems.

91影视

If \(A, B\) and \(C\) are sets, then \(A \times(B \cap C)=(A \times B) \cap(A \times C)\)

Short Answer

Step by step solution

Understanding The Terms

Prove the Left side Subset of Right side

Prove the Right side Subset of Left side

Key Concepts

Set theory

Intersection of sets

Subset

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Pure Maths

Mechanics Maths

Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.