What is the Symmetric Difference of 2 Sets?

[@robincondite]

Perfect and quick explanation! Thank you!

[@WrathofMath]

You're welcome, I am glad it helped! Let me know if there are ever any other lessons you'd like to see on the channel! Thanks for watching!

[@AtlehangClarkHlapisi]

simple and easy thanks!

[@WrathofMath]

You're welcome! Thanks for watching!

[@mathewabraham1294]

It really helped me.. Thankyou

[@WrathofMath]

+Mathew Abraham Thanks for watching Matthew, glad it helped!

[@siddhavinayakacharya5409]

sir if 2 sets does not having any common element than what will be the difference? is it an empty or null set

[@WrathofMath]

Thanks for watching and for the question! Let A and B be two sets with no common elements, meaning their intersection is empty. Remember that the symmetric difference is the set of all elements that are in A, or in B, but not in both! So let's give the symmetric difference of A and B a name, let's call it D. Thus, D has all elements that are in A or in B, but not in both. So what elements does it have? Remember that A and B have no elements in common!

[@azizhani33]

As per the explanation, 1) two sets with no common elements means their intersection is empty, and 2) the symmetric difference of two sets A and B is D = A ∆ B = (A union B) - (A intersect B). Therefore here D = (A union B) - Empty Set = A U B - {} = A U B.

[@legacyZon]

What's the difference between "difference of sets" and "symmetric difference of two sets" ?

[@WrathofMath]

Great question and thanks for watching! The difference of sets, set difference as it is often called, refers to set subtraction. So if A = { 1, 2, 3 } and B = { 2, 4, 5 }, then A - B, the set B subtracted from A, is equal to { 1, 3 }. We just take anything in A out if it is also in B. So since 2 is in A and B, it is not in A - B. So A - B has all elements of A that are NOT in B. Note also that A - B is often written as A\B, and A - B does not necessarily equal B - A, just like with subtraction on real numbers. So that is set difference, or set subtraction, which is not the same as the symmetric difference of sets. If we took the symmetric difference of A and B, we would have { 1, 3, 4, 5 }. This is the set that has everything in A, as well as everything in B, except for the elements that are in both A and B. Since 2 is in A and B, it is not in the symmetric difference of A and B. The symmetric difference of A and B is defined as (A - B) U (B - A). Looking at that definition, or at venn diagrams of symmetric difference, both help clarify why it is called "symmetric" difference. Does that make sense?

[@legacyZon]

@@WrathofMath So let's say we have to sets- A{1,2,3, 5} and B{1,2,3,4} Then difference of A-B = 5 and B-A = 4 And symmetric difference would be: 5 and 4? Right?!

[@WrathofMath]

Exactly! Just make sure you write A - B = { 5 } and B - A = { 4 } rather than A - B = 5 and B - A = 4, since they are sets, not numbers.

[@legacyZon]

@@WrathofMath oh okay thanks! Also, what happens when there's no difference? How do we write it?

[@WrathofMath]

Great question! We wouldn't want to run into problems when we subtract 5 from 5, for example, so it is a good thing we have 0. Similarly, if A = { 1, 2, 3 }, we don't want to run into problems when we evaluate A - A. Good thing we have the empty set! For any set S, S - S = { }. That is just the set containing no elements. It is also written as ∅. Similarly, S Δ S, the symmetric difference of S and S, is also equal to the empty set. EDIT: Also, I didn't mean to imply that only equal sets have the empty set as their difference. For example, if A = { 1, 2, 3 } and B = { 1, 2, 3, 4, 5 }, then A - B = { }. If A is a subset of B, then A - B will always be the empty set. This is not how symmetric difference works, however. Think about when the symmetric difference of two sets would be equal to the empty set, keeping in mind that the symmetric difference of A and B, denoted A Δ B, is equal to (A - B) U (B - A), and I am sure you can figure it out!

[@abdulrahmanawad2456]

can you please explain the same operations but in graph theory ?

[@Chris-jose]

Thanks ❤❤🎉

[@WrathofMath]

You're welcome 😊

[@JoMan-e8j]

is it the same even if its BΔA

[@tjstarr2960]

That is a great question! Usually, in everyday algebra, subtraction is not commutative, meaning you can't switch the order of the two operands (5-6 is NOT equal to 6-5) and get the same answer. But in this case, think of what BΔA means. It can be re-written as "(B ∪ A) - (B ∩ A)", as he stated in the video. Since the union and intersection operations are commutative, we know that (B ∪ A) = (A ∪ B) and also that (B ∩ A) = (A ∩ B), so we can say that (B ∪ A) - (B ∩ A) = (A ∪ B) - (A ∩ B). So, yes, the set difference is commutative, you can switch the order of the 2 sets and it doesn't matter, AΔB = BΔA

[@santhosh6594]

Good

[@WrathofMath]

Thank you! Let me know if you ever have any questions!