Question

Let \(\rho\) be a relation defined on a set of natural numbers N, as \(\rho\)={(x,y)∈N×N:2x+y=4}, then domain A and range B are 

• A. A⊂{x∈N:1≤x≤20} and B⊂{y∈N:1≤y≤39}
• B. A={x∈N:1≤x≤15} and B={y∈N:2≤y≤30}
• C. A≡N,B≡Q
• D. A=Q,B=Q

Answer 1

The Correct Option is A

Given: Relation ρ on natural numbers: $ρ = \{(x, y) \in \mathbb{N} \times \mathbb{N} : 2x + y = 4\}$ 

Step 1: We are looking for natural number solutions to the equation: $2x + y = 4$  
Rewriting: $y = 4 - 2x$ 

Step 2: Try natural number values of $x$:

• For $x = 1$: $y = 4 - 2(1) = 2$ → valid (since $y \in \mathbb{N}$)
• For $x = 2$: $y = 4 - 4 = 0$ → not valid (0 ∉ ℕ)
• For $x = 3$: $y = 4 - 6 = -2$ → not valid

So, only valid ordered pair is $(1, 2)$ 

Step 3: Now, find domain A and range B:

• Domain A: set of first elements = $\{1\}$
• Range B: set of second elements = $\{2\}$

These sets are subsets of:

• $A \subset \{x \in \mathbb{N} : 1 \le x \le 20\}$
• $B \subset \{y \in \mathbb{N} : 1 \le y \le 39\}$

Final Answer: (A): $A \subset \{x \in \mathbb{N} : 1 \le x \le 20\}, B \subset \{y \in \mathbb{N} : 1 \le y \le 39\}$