Prove that if X ⊆Y, then Y-(Y-X)=X for all sets X and Y. |...

Name: Problem 26, Chapter 2: Discrete Mathematics
Uploaded: 2022-12-01T03:59:01-08:00
Duration: 4 min 25 s
Channel: Diwakar Mandilwar
Description: Prove that if X ⊆Y, then Y-(Y-X)=X for all sets X and Y.

Key Concepts

Subset Relation

A subset relation (denoted by X ? Y) means that every element of set X is also an element of set Y. This concept is foundational in set theory, as it allows one to compare sets and understand inclusion properties.

Set Difference

The set difference, expressed as Y - X, is the collection of elements that are present in Y but not in X. This operation is essential in set theory for removing certain elements from a set to form a new set.

Set Equality and Double Inclusion Proof

Set equality is established by demonstrating that two sets have exactly the same elements, typically by proving each is a subset of the other. This method, known as proof by double inclusion, is fundamental when reasoning about the properties and relationships between sets.

Transcript

00:01 In this question, we have to prove the given expression.

00:03 So we can say that consider two sets, x and y, and they have the relation between them, such that x is a improper subset of y.

00:12 Now, recall that a minus b is equals to a intersection b bar, right? so from this very relation, we can say that the above sets can be written as y minus y minus of x and this will be equals to y intersection y minus x bar.

00:38 Right.

00:39 So from this very information we can here say that if it is simplified further, then it will be y minus and then it will be y minus x and that is equals to here.

00:50 Y intersection of y minus x then here it will be bar and then here we will have one more bar like this then here we can say that recall another relation which says that a intersection b and then here it is bar it will be equals to a bar union b bar so from this relation we can say that our expression will look like minus it will be y minus x and that will be equals to y intersection of y bar and then it will be union x double bar so from this information we can say that this will be equals to here as y intersection and then it will be y bar and then it is union x so now we can say that this has been written since here we know that x double bar is equals to x so from this very relation only we have written the above expression so using the distributive law to simplify further the above equation can be written as y minus y minus of x and that is equals to y intersection of y bar and then here it will be union x so in this way we can say that the above expression will be over here so basically we have here use the distribution law, right? so we can write it over here and it will be distributive law, not distribution law, very sorry for saying it wrong.

02:30 It will be distributive law.

02:32 Now here it will be say y minus y minus x and that will be equals to here y intersection y bar and then here we are going to have union y and then it will be intersection.

02:50 And then here it will be x.

02:52 Now we can say that it is going to be over here, y minus y minus x and that will be equals to phi and then it is union y intersection x...

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