Problem 17 For all sets $A, B$, and $C$... [FREE SOLUTION]

Chapter 5: Problem 17

For all sets $A, B$, and $C$ $$ A \times(B \cap C)=(A \times B) \cap(A \times C) $$

Short Answer

Expert verified

To prove that $A \times (B \cap C) = (A \times B) \cap (A \times C)$, we showed that every element of $A \times (B \cap C)$ is an element of $(A \times B) \cap (A \times C)$ and vice versa. By analyzing the properties of Cartesian products and intersections, we established that for any arbitrary element $(a, x)$ in either set, the element would also belong to the other set. Thus, the equality $A \times(B \cap C)=(A \times B) \cap(A \times C)$ holds true.

Step by step solution

Proving Elements of $A \times (B \cap C)$ are in $(A \times B) \cap (A \times C)$

Let $(a, x)$ be an arbitrary element in $A \times (B \cap C)$. Since $(a, x) \in A \times (B \cap C)$, this means that $a \in A$ and $x \in (B \cap C)$. Since $x \in (B \cap C)$, we have that x is an element of both B and C, i.e., $x \in B$ and $x \in C$. Now, because $a \in A$ and $x \in B$, we have $(a, x) \in A \times B$. Similarly, because $a \in A$ and $x \in C$, we have $(a, x) \in A \times C$. Thus, $(a, x) \in (A \times B) \cap (A \times C)$, since it is in both $A \times B$ and $A \times C$.

Proving Elements of $(A \times B) \cap (A \times C)$ are in $A \times (B \cap C)$

Let $(a, x)$ be an arbitrary element in $(A \times B) \cap (A \times C)$. Since $(a, x) \in (A \times B) \cap (A \times C)$, this means that $(a, x) \in A \times B$ and $(a, x) \in A \times C$. Because $(a, x) \in A \times B$, we have that $a \in A$ and $x \in B$. Similarly, because $(a, x) \in A \times C$, we have that $a \in A$ and $x \in C$. Since $x \in B$ and $x \in C$, we have that $x \in (B \cap C)$. And because $a \in A$ and $x \in (B \cap C)$, we have that $(a, x) \in A \times (B \cap C)$. We have shown that every element of $A \times (B \cap C)$ is an element of $(A \times B) \cap (A \times C)$ and vice versa. Therefore, we have proven that: $$ A \times(B \cap C)=(A \times B) \cap(A \times C) $$

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Key Concepts

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

Cartesian Product

The Cartesian product is a mathematical operation that returns a new set from two given sets. If you have a set $A$ and a set $B$, their Cartesian product is noted as $A \times B$. It consists of all possible ordered pairs $(a, b)$ such that $a$ is an element of $A$ and $b$ is an element of $B$.
For example, if $A = \{1, 2\}$ and $B = \{x, y\}$, then the Cartesian product $A \times B = \{(1, x), (1, y), (2, x), (2, y)\}$.

Each pair starts with an element from $A$ and ends with an element from $B$.
The order of the pairs matters, so $(a, b)$ is different from $(b, a)$.

Cartesian products are fundamental in understanding relations and functions in set theory. They form the basis for exploring more complex operations like intersections and unions between these products.

Intersection of Sets

The intersection of sets is a foundational concept in set theory. When you find the intersection of two sets $B$ and $C$, you get a new set that consists of all the elements common to both $B$ and $C$.
This is denoted as $B \cap C$. For instance, if $B = \{1, 2, 3\}$ and $C = \{2, 3, 4\}$, then $B \cap C = \{2, 3\}$.

Only elements appearing in both sets are included in the intersection.
If there are no common elements, the intersection is an empty set, denoted $\emptyset$.

In mathematical operations, intersecting sets within a product like $A \times (B \cap C)$ allows you to focus on only those ordered pairs where the second component belongs to both sets $B$ and $C$.
The intersection helps in narrowing down possibilities and is extensively used in proofs and logical deductions.

Element Proof

Element proof involves showing that a certain element belongs to a particular set. In our exercise, proof is used to show equality between $A \times (B \cap C)$ and $(A \times B) \cap (A \times C)$ by demonstrating that any element $(a, x)$ in one set is also in the other and vice versa.
To do this effectively, you pursue two paths:

First, pick an arbitrary element from $A \times (B \cap C)$ and verify it belongs to $(A \times B) \cap (A \times C)$.
Secondly, take an arbitrary element from $(A \times B) \cap (A \times C)$ and show it belongs to $A \times (B \cap C)$.

This step-by-step approach establishes that every element in one set is mirrored exactly in the other.
By resolving both directions of the proof, we reinforce the logical equivalence of the sets, emphasizing a core principle of mathematical logic: equality of sets through mutual inclusion.

Mathematical Logic

Mathematical logic provides the underlying framework for doing proofs and reasoning in mathematics. In this context, it tells us how to systematically show that two expressions represent the same set.
Using logic helps us logically connect all steps in proving the set identity $A \times (B \cap C) = (A \times B) \cap (A \times C)$.
Here are key principles employed:

Logical Deduction: Derive conclusions based on known premises. If $x$ is in $B \cap C$, then $x$ must be in both $B$ and $C$.
Contraposition: Understanding that if $P \Rightarrow Q$ is true, then $eg Q \Rightarrow eg P$ is also true.
Bi-conditional Statements: Showing both $P \Rightarrow Q$ and $Q \Rightarrow P$ proves $P \iff Q$.

By applying these logical tools, mathematicians can validate formal proofs and confirm conclusions, such as proving two sets are identical using set operations.

91影视

For all sets \(A, B\), and \(C\) $$ A \times(B \cap C)=(A \times B) \cap(A \times C) $$

Short Answer

Step by step solution

Proving Elements of \(A \times (B \cap C)\) are in \((A \times B) \cap (A \times C)\)

Proving Elements of \((A \times B) \cap (A \times C)\) are in \(A \times (B \cap C)\)

Key Concepts

Cartesian Product

Intersection of Sets

Element Proof

Mathematical Logic

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