- #1
ainster31
- 158
- 1
Homework Statement
Homework Equations
I have to use these set identities:
The Attempt at a Solution
Pretty sure this is impossible because there's no identity for the Cartesian product.
Set Identity Proofs: Exploring the Cartesian Product
In summary, the task at hand is to use set identities to show that A X (B ∪ C) is equal to the union of (A X B) and (A X C). This can be done by considering the elements in the Cartesian product and showing that they are also in the union, and vice versa.
I have to use these set identities:
Pretty sure this is impossible because there's no identity for the Cartesian product.
- #2
kduna
- 52
- 7
Just go at it the old fashion way.
Suppose (a, d) [itex]\in[/itex] A X (B [itex]\cup[/itex] C). Then a [itex]\in[/itex] A. Also d [itex]\in[/itex] B or d [itex]\in[/itex] C. So (a,d) [itex]\in[/itex] (A X B) or (a,d) [itex]\in[/itex] (A X C).
Thus (a,d) [itex]\in[/itex] (A X B) [itex]\cup[/itex] (A X C).
Therefore A X (B [itex]\cup[/itex] C) [itex]\subseteq[/itex] (A X B) [itex]\cup[/itex] (A X C).
Proving the subset goes the other way follows similarly.
Suppose (a, d) [itex]\in[/itex] A X (B [itex]\cup[/itex] C). Then a [itex]\in[/itex] A. Also d [itex]\in[/itex] B or d [itex]\in[/itex] C. So (a,d) [itex]\in[/itex] (A X B) or (a,d) [itex]\in[/itex] (A X C).
Thus (a,d) [itex]\in[/itex] (A X B) [itex]\cup[/itex] (A X C).
Therefore A X (B [itex]\cup[/itex] C) [itex]\subseteq[/itex] (A X B) [itex]\cup[/itex] (A X C).
Proving the subset goes the other way follows similarly.
Related to Set Identity Proofs: Exploring the Cartesian Product
1. What is a Cartesian product?
A Cartesian product is a mathematical operation that combines two sets to create a new set consisting of all possible ordered pairs of elements from the two original sets. It is represented by the symbol x and is often used in algebra, geometry, and computer science to model relationships between two sets.
2. How is a Cartesian product related to set identity proofs?
Set identity proofs use the Cartesian product to explore the relationship between two sets and determine whether they are equal or not. By taking the Cartesian product of two sets and comparing the resulting sets, one can prove that they are either identical or not.
3. What is the process of exploring a Cartesian product in set identity proofs?
The process starts by taking the Cartesian product of two sets. Then, the resulting set is compared to the original sets to see if they contain the same elements. If they do, then the two sets are proven to be equal. If not, then the two sets are not equal.
4. What are some real-life applications of set identity proofs and the Cartesian product?
Set identity proofs and the Cartesian product are commonly used in computer science and programming to determine if two sets of data are identical. They are also used in mathematical modeling and data analysis to explore relationships between different sets of data.
5. Can the Cartesian product be applied to more than two sets?
Yes, the Cartesian product can be applied to any number of sets. The resulting set will contain all possible combinations of elements from the original sets. For example, the Cartesian product of three sets A, B, and C would result in a set containing ordered triples (a, b, c) where a is an element of A, b is an element of B, and c is an element of C.
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