1. Basic Idea Behind These Properties
The Cartesian product behaves in predictable ways when sets are swapped, combined, or restricted. These properties help understand how ordered pairs form and how products change when the sets change.
2. Order Matters
Swapping the sets usually gives a different product because the positions of the elements in each ordered pair change.
2.1. Property
A \times B \neq B \times A \quad \text{(in general)}
2.2. Example
A = \{1,2\},\ B = \{3\}
A \times B = \{(1,3),(2,3)\}
B \times A = \{(3,1),(3,2)\}
3. Cartesian Product With the Empty Set
If any set is empty, there are no elements to pair with the other set, so the product is empty.
3.1. Property
A \times \emptyset = \emptyset \times A = \emptyset
3.2. Example
A = \{1,2\},\ \emptyset = \emptyset
A \times \emptyset = \emptyset
4. Products With Subsets
If one set is a subset of another, its product with a third set is also a subset of the corresponding larger product.
4.1. Property
A_1 \subseteq A_2 \Rightarrow A_1 \times B \subseteq A_2 \times B
4.2. Example
A_1 = \{1\},\ A_2 = \{1,2\},\ B = \{x,y\}
A_1 \times B = \{(1,x),(1,y)\}
A_2 \times B = \{(1,x),(1,y),(2,x),(2,y)\}
5. Distributive Behaviour Over Unions
The Cartesian product distributes over union. This helps simplify products when sets are combined.
5.1. Property
A \times (B \cup C) = (A \times B) \cup (A \times C)
5.2. Example
A = \{1\},\ B = \{2\},\ C = \{3\}
A \times (B \cup C) = \{(1,2),(1,3)\}
(A \times B) \cup (A \times C) = \{(1,2),(1,3)\}
6. Distributive Behaviour Over Intersection
The product also distributes over intersection.
6.1. Property
A \times (B \cap C) = (A \times B) \cap (A \times C)
6.2. Example
A = \{1\},\ B = \{2,3\},\ C = \{3,4\}
B \cap C = \{3\}
A \times (B \cap C) = \{(1,3)\}
(A \times B) \cap (A \times C) = \{(1,3)\}
7. Non-Associativity
Cartesian products are not strictly associative because ordered pairs cannot be grouped freely. The structure of the pairs changes.
7.1. Illustration
(A \times B) \times C = \{((a,b),c)\}
A \times (B \times C) = \{(a,(b,c))\}
These have different internal structure, so they are not equal.
8. Important Notes
Some quick points to remember:
8.1. Key Points
- Order of sets matters.
- If any set is empty, the product is empty.
- Subset relations extend naturally in Cartesian products.
- Products distribute over union and intersection.
- Not associative because of the structure of pairs.