Questions: T = (x, y) x ∈ ℕ and y = 2x

T = (x, y) x ∈ ℕ and y = 2x
Transcript text: \[ T=\{(x, y) \mid x \in \mathbb{N} \text { and } y=2 x\} \]
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Solution

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Solution Steps

To find the range of the given relation \( T = \{(x, y) \mid x \in \mathbb{N} \text{ and } y = 2x\} \), we need to determine the set of all possible values of \( y \) when \( x \) is a natural number. Since \( y = 2x \), the range will be all even natural numbers.

Step 1: Define the Relation

The relation \( T \) is defined as \( T = \{(x, y) \mid x \in \mathbb{N} \text{ and } y = 2x\} \). Here, \( x \) represents natural numbers, and \( y \) is determined by the equation \( y = 2x \).

Step 2: Determine the Range

To find the range of the relation, we calculate \( y \) for the first few natural numbers \( x \). For \( x = 1, 2, \ldots, 10 \), the corresponding values of \( y \) are:

  • For \( x = 1 \): \( y = 2 \times 1 = 2 \)
  • For \( x = 2 \): \( y = 2 \times 2 = 4 \)
  • For \( x = 3 \): \( y = 2 \times 3 = 6 \)
  • For \( x = 4 \): \( y = 2 \times 4 = 8 \)
  • For \( x = 5 \): \( y = 2 \times 5 = 10 \)
  • For \( x = 6 \): \( y = 2 \times 6 = 12 \)
  • For \( x = 7 \): \( y = 2 \times 7 = 14 \)
  • For \( x = 8 \): \( y = 2 \times 8 = 16 \)
  • For \( x = 9 \): \( y = 2 \times 9 = 18 \)
  • For \( x = 10 \): \( y = 2 \times 10 = 20 \)

Thus, the range of \( T \) for these values of \( x \) is \( \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\} \).

Step 3: Generalize the Range

Since \( x \) can take any natural number, the general form of \( y \) is \( y = 2x \) where \( x \in \mathbb{N} \). Therefore, the range of the relation \( T \) is all even natural numbers, which can be expressed as \( \{y \in \mathbb{N} \mid y \text{ is even}\} \).

Final Answer

The range of the relation \( T \) is all even natural numbers, which can be expressed as \( \boxed{\{y \in \mathbb{N} \mid y \text{ is even}\}} \).

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