Question

Let a, b, u, and v be any sets such that a ⊆ u and b ⊆ v . prove or disprove a × b ⊆ u × v .

Answer 1

To prove or disprove the statement a × b ⊆ u × v, we can use the definition of the Cartesian product and the given set inclusions to show that if an element is in a × b, it is also in u × v.

Step-by-step explanation:

To prove or disprove the statement a × b ⊆ u × v:

1. Let (x,y) be an arbitrary element in a × b.
2. By definition, this means that x is an element of a and y is an element of b.
3. Since a ⊆ u and b ⊆ v, we can conclude that x is also an element of u and y is also an element of v.
4. Therefore, (x,y) is an element of u × v.
5. Since (x,y) was an arbitrary element, we can conclude that a × b ⊆ u × v.

Example: If a = {1,2} and b = {3,4}, and u = {1,2,5} and v = {3,4,6}, then a × b = {(1,3),(1,4),(2,3),(2,4)} and u × v = {(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(5,3),(5,4),(5,6)}.