Cartesian product and symmetric difference

In summary, we are trying to prove that for any element p in the set (A x B) △ (A x C), it is also in the set A x (B △ C). We show this by breaking down the elements of p and using the properties of the symmetric difference. By doing so, we can see that Ax(BΔC) is a subset of (AxB) Δ (AxC), therefore proving the equality between the two sets.
  • #1
fatineouahbi
10
0
Let A,B,C be three sets . Prove Ax(BΔC)= (AxB) Δ (AxC)

I tried to start with this :

Let p be an arbitrary element of Ax(BΔC)
then p=(x,y) such that x ∈ A and y ∈ (BΔC)
x ∈ A and (y∈ B\C or y∈ C\B)
(x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B)

But I don't know how to continue or if I should even start with this .
 
Mathematics news on Phys.org
  • #2
fatineouahbi said:
Let A,B,C be three sets . Prove Ax(BΔC)= (AxB) Δ (AxC)

I tried to start with this :

Let p be an arbitrary element of Ax(BΔC)
then p=(x,y) such that x ∈ A and y ∈ (BΔC)
x ∈ A and (y∈ B\C or y∈ C\B)
(x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B)

But I don't know how to continue or if I should even start with this .

It is right so far.When does it hold that $p \in (A \times B) \triangle (A \times C)$ ?
 
  • #3
evinda said:
It is right so far.When does it hold that $p \in (A \times B) \triangle (A \times C)$ ?

Hello :) Thank you , I think I may get it now ?

(x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B)
then p ∈ Ax(B\C) or p ∈ Ax(C\B)
then p ∈ (AxB) \ (AxC) or p ∈ (AxC) \ (AxB)
thus p ∈ (AxB) △ (AxC)
then Ax(BΔC) ‎⊂ (AxB) Δ (AxC)

Then I'll just try to go backwards maybe ?
 
  • #4
fatineouahbi said:
Hello :) Thank you , I think I may get it now ?

(x ∈ A and y ∈ B\C) or (x ∈ A and y ∈ C\B)
then p ∈ Ax(B\C) or p ∈ Ax(C\B)
then p ∈ (AxB) \ (AxC) or p ∈ (AxC) \ (AxB)
thus p ∈ (AxB) △ (AxC)
then Ax(BΔC) ‎⊂ (AxB) Δ (AxC)
Well done, you are right :)

fatineouahbi said:
Then I'll just try to go backwards maybe ?
Yes, you pick an element in $(A \times B)\triangle (A \times C)$ and you need to show that it is also in $A \times (B \triangle C)$.
 
  • #5
evinda said:
Well done, you are right :)

Yes, you pick an element in $(A \times B)\triangle (A \times C)$ and you need to show that it is also in $A \times (B \triangle C)$.

Thank you so much !
 

Related to Cartesian product and symmetric difference

1. What is a Cartesian product?

A Cartesian product is a mathematical operation that combines two sets of elements to form a new set, where each element in the new set is a pair of elements from the original sets. It is denoted by the symbol "x" or by the word "cross". For example, if set A contains the elements {1, 2} and set B contains the elements {a, b}, the Cartesian product of A and B would be {(1, a), (1, b), (2, a), (2, b)}.

2. How is a Cartesian product different from a regular product?

A regular product is a multiplication operation between two numbers or variables, while a Cartesian product is an operation between two sets. In a regular product, the result is a single value, while in a Cartesian product, the result is a set of ordered pairs or tuples.

3. What is a symmetric difference?

A symmetric difference is a mathematical operation that compares two sets and returns a new set containing elements that are in either of the original sets, but not in both. It is denoted by the symbol "Δ" or by the phrase "symmetric difference". For example, if set A contains the elements {1, 2, 3} and set B contains the elements {2, 3, 4}, the symmetric difference of A and B would be {1, 4}.

4. How is a symmetric difference different from a regular difference?

A regular difference, also known as set difference, is a mathematical operation that compares two sets and returns a new set containing elements that are in the first set but not in the second set. In contrast, a symmetric difference compares two sets and returns a new set containing elements that are in either of the original sets, but not in both.

5. What is the relationship between Cartesian product and symmetric difference?

The Cartesian product and symmetric difference are two different mathematical operations that can be applied to sets. The Cartesian product combines elements from two sets to form a new set, while the symmetric difference compares elements from two sets to form a new set. They are related in that both operations involve combining or comparing elements from two sets to form a new set, but they serve different purposes and produce different results.

Similar threads

B How to prove the associative law of symmetric difference?
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
2K
I Partial Differential Equation solved using Products
    • General Math
Replies
2
Views
343
B Reprise: Calculate the distance between two points without using a coordinate system
    • General Math
Replies
1
Views
337
B Simple Combo/Permute calculation
    • General Math
2
Replies
45
Views
3K
B Invariant under rotation: Banal, obvious, or noteworthy?
    • General Math
Replies
2
Views
1K
B Are Inverse Problems More Complex Than Direct Problems in Mathematics?
    • General Math
Replies
13
Views
1K
MHB Volume of $E$ in $\mathbb{R}^4$ - AMM
    • General Math
Replies
1
Views
1K
I Bloch Analysis proof of Theorem 2.5.5 (Definition by recursion)
    • Topology and Analysis
Replies
2
Views
373
Can you use Taylor Series with mathematical objects other than points?
    • General Math
Replies
4
Views
1K
MHB How do I solve a system of sets ?
    • General Math
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top