Let \( R⊂ N × N \). Which of the... - Solved

Question 5

Mathematics Function and Relation Medium

Let \( R\subset\mathbb{N}\times\mathbb{N} \). Which of the following statement is necessarily true?

(A) If for each \( a\in\mathbb{N} \) the set \( R_{a}:=\{b\in\mathbb{N}:(a,b)\in R\} \) has cardinality at most 1, then R represents a function \( f:\mathbb{N}\rightarrow\mathbb{N} \)

(B) If for each \( b\in\mathbb{N} \) the set \( R^{b}:=\{a\in\mathbb{N}:(a,b)\in R\} \) has cardinality at most 1, then R represents a function \( f:\mathbb{N}\rightarrow\mathbb{N} \)

(C) If for some \( a\in\mathbb{N} \) the set \( R_{a} \) is infinite then \( R_{a} \) and R have the same cardinality

(D) If for all \( a\in\mathbb{N} \) the set \( R_{a} \) is infinite then the cardinality of R is larger than that of \( R_{a} \)

View Dynamic Solution & Explanation ▼

Correct Solution: Option C

Step-by-step Solution:

Let's analyze each option: Option A: Having cardinality at most 1 means some elements in the domain may have 0 mappings. This represents a partial function, not necessarily a total function \( f:\mathbb{N}\rightarrow\mathbb{N} \). Option B: This defines an injective property or partial inverse function, not a function from \( \mathbb{N} \) to \( \mathbb{N} \). Option C: If \( R_a \) is infinite, since \( R_a \subset \mathbb{N} \), it must be countably infinite (cardinality \( \aleph_0 \)). Since \( R \subset \mathbb{N} \times \mathbb{N} \), \( R \) is at most countably infinite. Because \( R_a \times \{a\} \subseteq R \), \( R \) must also be at least countably infinite. Thus, both \( R_a \) and \( R \) have the exact same cardinality (\( \aleph_0 \)). This is necessarily true. Option D: Both would be countably infinite, meaning their cardinalities are equal, not larger.