In the realm of mathematics, the concept of infinity and the cardinality of infinite sets often lead to fascinating insights and profound proofs. One such proof involves demonstrating that there is no one-to-one correspondence (bijection) between the set of real numbers R\mathbb{R}R and the set of rational numbers Q\mathbb{Q}Q. This proof, famously known as Cantor’s diagonal argument, serves as a cornerstone in set theory, illustrating the uncountability of real numbers compared to the countability of rational numbers.

Assumptions and Definitions

Let’s begin by assuming the existence of a bijection f:R→Qf: \mathbb{R} \to \mathbb{Q}f:R→Q. This function fff would map every real number to a unique rational number and vice versa, implying a perfect pairing between these two infinite sets.

Cantor’s Diagonal Argument

To demonstrate the impossibility of such a bijection, we construct a real number rrr that cannot be included in the image of fff.

1. Listing Rational Numbers: Start by listing all rational numbers {q1,q2,q3,…}\{ q_1, q_2, q_3, \ldots \}{q1​,q2​,q3​,…}, where each qiq_iqi​ corresponds to a real number ri=f−1(qi)r_i = f^{-1}(q_i)ri​=f−1(qi​).
2. Constructing a New Real Number: Formulate a new real number rrr by altering the iii-th digit of the decimal expansion of rir_iri​:
   • If r1=0.a11a12a13…r_1 = 0.a_{11}a_{12}a_{13}\ldotsr1​=0.a11​a12​a13​…
   • If r2=0.a21a22a23…r_2 = 0.a_{21}a_{22}a_{23}\ldotsr2​=0.a21​a22​a23​…
   • If r3=0.a31a32a33…r_3 = 0.a_{31}a_{32}a_{33}\ldotsr3​=0.a31​a32​a33​…
   Construct r=0.b1b2b3…r = 0.b_1b_2b_3\ldotsr=0.b1​b2​b3​…, where each bi≠aiib_i \neq a_{ii}bi​=aii​.
3. Difference from rir_iri​: By construction, rrr differs from each rir_iri​ in at least the iii-th decimal place.
4. Contradiction: Since rrr is constructed to differ from every rir_iri​, it follows that rrr cannot be in the image of fff. This contradicts the assumption that fff is a bijection.

Conclusion

Therefore, we conclude that there is no bijection f:R→Qf: \mathbb{R} \to \mathbb{Q}f:R→Q. This means there is no one-to-one correspondence between the set of real numbers R\mathbb{R}R and the set of rational numbers Q\mathbb{Q}Q. The proof hinges on Cantor’s diagonal argument, which elegantly shows that the real numbers are uncountably infinite, while the rational numbers are countably infinite.

This fundamental result underscores the richness and complexity of infinite sets in mathematics, revealing deep insights into the structure of numbers and the nature of infinity itself.