NCERT Class 12 · Ch 1 Exercise 1.2 · Q8
NCERT Class 12 Maths · Chapter 1 Relations And Functions · Exercise 1.2 Question 8
For sets A,B, prove that f:A×B→B×A, f(a,b)=(b,a), is bijective.
💡 Concept Hints
- Use equality of ordered pairs, then reverse the swap for an arbitrary target pair.
- Separate the one-one and onto claims. For onto, ensure each inverse candidate lies in the allowed domain.
✅ Step-by-Step Solution
Final Answer: f is bijective, including when a Cartesian product is empty.
- If f(a,b)=f(c,d), then (b,a)=(d,c). Equality of ordered pairs yields b=d and a=c, so the original pairs are equal; f is injective.
- Given any (u,v)∈B×A, the pair (v,u)∈A×B satisfies f(v,u)=(u,v), proving onto.
- Swapping coordinates again is the inverse map. If either set is empty, both products are empty and the same injectivity and surjectivity conditions hold vacuously.
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