Problem 5 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(R\) be a relation defined on the set of natural numbers \(N\) as \(R=[(x, y): x \in N, y \in N, 2 x+y=41]\). Then (A) Domain of \(R=\\{1,2,3, \ldots, 19,20\\}\) (B) Range of \(R=\\{39,37,35,9,7,5,3,1\\}\) (C) \(R\) is reflexive (D) \(R\) is symmetric

Step-by-Step Solution

Answer

(A) True; (B) True; (C) False; (D) False.

Step 1: Understand the Relation

The given relation \(R\) is defined as the set of all ordered pairs \((x, y)\) for which \(2x + y = 41\). This means for each value of \(x\), there is a corresponding \(y\) such that this equation holds true.

Step 2: Determine the Domain

The domain of a relation is the set of all first elements \(x\) from the ordered pairs. For the equation \(2x + y = 41\), we need to find all possible natural numbers \(x\) such that after choosing \(x\), \(y\) also remains a natural number. Solving for \(x\): \(y = 41 - 2x\). Since \(y\) is a natural number, it should be non-negative: \(41 - 2x > 0\), simplifying to \(x < 20.5\). Thus, \(x\) can be any natural number from \(1\) to \(20\). So, the domain is \(\{1, 2, 3, \ldots, 19, 20\}\).

Step 3: Determine the Range

The range is the set of all second elements \(y\) from the ordered pairs. For each allowable \(x\) calculated previously, \(y = 41 - 2x\). Calculate \(y\) for each \(x\) from 1 to 20. You will find that for \(x = 1, y = 39\); \(x = 2, y = 37\); \(x = 3, y = 35\); and so on, until \(x = 20, y = 1\). The different values of \(y\) form the range: \(\{39, 37, 35, \ldots, 9, 7, 5, 3, 1\}\).

Step 4: Check for Reflexivity

A relation is reflexive if for all elements \(a\) in the domain, the pair \((a, a)\) is in the relation. For \(R\), we need \(2a + a = 41\), implying \(3a = 41\). Since \(41/3\) is not a natural number, \(R\) cannot be reflexive as no natural number \(a\) satisfies the condition \((a, a) \in R\).

Step 5: Check for Symmetry

A relation is symmetric if whenever \((a, b) \in R\), then \((b, a) \in R\) as well. For \(R\), assume \((x, y) \in R\) which means \(2x+y = 41\). For \((y, x)\) to be in \(R\), we must have \(2y+x = 41\). If both conditions happen, then it implies: \(2x+y = 2y+x\), simplifying to \(x = y\), which is a contradiction per our findings that \(xeq y\). Hence, \(R\) is not symmetric.

Key Concepts

Domain and Range

In mathematics, understanding the concepts of domain and range is very crucial when dealing with relations. Let's start with the **domain**. The domain comprises all possible input values or the first elements of the ordered pairs involved in a relation. For the relation defined on natural numbers by the equation \(2x + y = 41\):
\[\]

• We rearrange for \(y\) to give \(y = 41 - 2x\).
• For \(y\) to remain a natural number, \(41 - 2x\) must be greater than zero, limiting \(x\) to natural numbers less than 20.5.
• The possible values of \(x\) range from 1 to 20, forming the domain \(\{1, 2, 3, \ldots, 20\}\).

To find the **range**, consider the potential outputs or second elements \(y\) as determined by the earlier rearrangement. For each \(x\) from the domain:
\[\]

• Substitute into \(y = 41 - 2x\) to find values of \(y\).
• As \(x\) increases from 1 to 20, \(y\) decreases from 39 to 1, forming the range \(\{39, 37, 35, \ldots, 1\}\).

Reflexivity

The property of reflexivity in relations is about elements relating to themselves. For a relation to be reflexive, every element in its domain should be able to pair with itself in the set. In mathematical terms, if \(a\) is an element from the domain, the pair \((a, a)\) must exist in the relation. Let's apply this to our example relation, \(R\):
\[\]

• Given the condition of our relation \(2x + y = 41\), for reflexivity, substitute \(y = x\), leading to \((2x + x) = 41\).
• This equation simplifies to \(3x = 41\).
• Solving for \(x\), we find \(x = \frac{41}{3}\), which is not a natural number.

Since no natural number satisfies \((x, x)\) in this context, the relation \(R\) is not reflexive.

Symmetry

Symmetry in relations is about mutual relationships. A relation is symmetric if whenever \((a, b)\) is in the relation, \((b, a)\) should be included as well. To evaluate symmetry for our relation \(R\), consider any pair \((x, y)\) belonging to \(R\):
\[\]

• This means the pair satisfies \(2x + y = 41\).
• If \(R\) is symmetric, \(2y + x = 41\) should also hold, for \((y, x)\) to be in \(R\).
• Reworking the conditions implies: \(2x + y = 2y + x\) which simplifies to \(x = y\).

However, analysis reveals that \(x\) does not equal \(y\) for any of the solutions found, hence \(R\) is not symmetric. Symmetry fails as none of the pairs reverse to substitute into the original condition, affirming that \(R\) isn't a symmetric relation.