Cartesian Products Of Sets

Concept: For finite sets, the number of elements in the Cartesian product is:

Step 1: Form equations using given data.


Step 2: Multiply all three equations.


Step 3: Take square root.


Step 4: Divide (1) by (2).


Step 5: Substitute into (3).

Concept:
• A function is one-to-one (injective) if distinct elements of domain have distinct images.
• Total number of elements in domain must be number of elements in codomain for injectivity.
• If , injective mapping is impossible (Pigeonhole Principle).

Step 1: Check cardinality of sets


Step 2: Apply Pigeonhole Principle
Since there are more elements in than in , at least two elements of must map to the same element in .
Hence, injective (one-to-one) mapping is not possible.

Step 3: Verify options individually
Option (A): Missing element not a function.
Option (B): and both map to not one-to-one.
Option (C): Missing element not a function.
Option (D): and both map to not one-to-one.
Option (E): Correct, as one-to-one function is not possible. Final Conclusion:
One-to-one function from to cannot be defined.