Prove or disprove that for all sets A, B, and C, we have a) A ×(B ∪C)=(A ×B) ∪(A ×C) b) A ×(B ∩C

Prove or disprove that for all sets A, B, and C, we have a)...

Key Concepts

Cartesian Product

The Cartesian product of two sets is the set of all ordered pairs where the first component comes from the first set and the second component comes from the second set. This concept is fundamental to creating product spaces and exploring multidimensional relationships between sets.

Set Union

The union of two sets is the set that contains all elements that are in either one of the sets or in both. It is a basic operation in set theory that combines the elements of two sets without duplication.

Set Intersection

The intersection of two sets is the set that consists of only those elements that are common to both sets. This operation identifies shared elements and is core to understanding overlapping properties of sets.

Distributive Properties in Set Theory

Distributive properties describe how one set operation interacts with another. Specifically, the Cartesian product distributes over the union and intersection operations, meaning that taking the product of a set with a union (or intersection) of sets yields the same result as the union (or intersection) of the Cartesian products. This principle is key when proving equivalences in set operations.

Transcript

00:01 Okay, given three sets, a, b, c, we want to see some relationship between the cartesian product.

00:10 I mean, the cartesian product of a with b union c.

00:15 We want to show it is actually equal to the cartesian product of a cross b union a cross c.

00:22 Okay, notice b is a subset of b union c.

00:31 That means if we do the cartesian product on the left side, this guy must be a subset of the right side.

00:45 The same as c is a subset of b union c again.

00:50 And if we do the cartesian product, we get the same thing.

00:54 That means those two sets are both a subset of the left one.

01:05 That means the union of them, both of them are a subset of the left one.

01:13 That means the union of them must be a subset of the left one.

01:21 So we have proved this condition, this relationship.

01:26 Now let's consider this guy.

01:31 Okay, for any element that x is contained in this quotation product, by the definition of the quotation product, we know x can be written as a and some, let's say d, where a is in capital a and d is contained in this union.

01:55 Union.

01:57 As d is contained in this union, we know we have two cases.

02:00 We have two things to discuss first.

02:03 A is contained in a, and d is contained in b.

02:09 That means a d, which is equal to x, must be an subset, must be an element in this guy.

02:20 This is a subset of a cross b union a cross c.

02:27 For the second one, little a is containing capital a, and d is containing c.

02:35 From the same discussion, we know that now a d, which is equal to x, is an element in a cross c, which is a subset of a cross b union a cross c.

02:47 So for any element in this set, we prove x must be an element in this guy in two cases.

03:03 In both of those two cases, that means x is indeed an element in this set...

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