Question

A Relation \(R\) is defined in the set of integers I as follows \((\mathrm{x}, \mathrm{y}) \in \mathrm{R}\) if \(\mathrm{x}^{2}+\mathrm{y}^{2}=9\) which of the following is false ? (a) \(\mathrm{R}=\\{(0,3),(0,-3),(3,0),(-3,0)\\}\) (b) Domain of \(\mathrm{R}=\\{-3,0,3\\}\) (c) Range of \(\mathrm{R}=\\{-3,0,3\\}\) (d) None of the above

Short answer

The false statement among the given options is (d) None of the above, as all other statements (a), (b), and (c) are true according to our step-by-step analysis.

Step-by-step solution

Evaluate statement (a)

Check if R = {(0,3),(0,-3),(3,0),(-3,0)} by satisfying the given condition x^2 + y^2 = 9. If we plug in the values of x and y from the pairs given in statement (a) into the equation x^2 + y^2, we get: - (0,3) => 0^2 + 3^2 = 0 + 9 = 9 - (0,-3) => 0^2 + (-3)^2 = 0 + 9 = 9 - (3,0) => 3^2 + 0^2 = 9 + 0 = 9 - (-3,0) => (-3)^2 + 0^2 = 9 + 0 = 9 All the pairs satisfy the given relation, so statement (a) is true.

Evaluate statement (b)

Check if the domain of R is {-3,0,3}. Domain is the set of all x values such that (x, y) is in R. From the pairs obtained in step 1, we can see that the x values are -3, 0, and 3, which do belong to the integer set. Therefore, statement (b) is true.

Evaluate statement (c)

Check if the range of R is {-3,0,3}. Range is the set of all y values such that (x, y) is in R. From the pairs obtained in step 1, we can see that the y values are -3, 0, and 3, which do belong to the integer set. Therefore, statement (c) is true.

Determine the false statement

Since statements (a), (b), and (c) are all true, the false statement among the given options is (d) None of the above.

Key concepts

Domain and Range of a Relation

Understanding the domain and range of a relation is fundamental in mathematics. It's like knowing the guests expected at a party (domain) and those who actually turned up (range).

Let's imagine a relation as a machine that takes input and produces output; the set of all possible inputs is called the 'domain', and the set of all actual outputs is known as the 'range'. In our exercise, we're hosting a mathematical party with the integer set \(I\), and we only invite pairs where the sum of their squares is 9.

When evaluating which guest pairs can come to the party, we see that for any pair \( (x, y) \) in the relation \( R \), the domain consists of the 'x' values. Since we have \( -3, 0, \text{and} 3 \) as our 'x' values, they form the guest list or the domain. Meanwhile, the range, our actual party attendees, are the 'y' values \( -3, 0, \text{and} 3 \) derived from the relation's condition \( x^2 + y^2 = 9 \).

It's key to note that the domain and range must consist of values that satisfy the given relation and belong to the specified set, in this case, the integers.

Integer Set Relations

An integer set relation is a connection between pairs of integers that satisfy a specific condition. Imagine these as special handshakes only certain pairs can accomplish based on a secret code. In our problem, the secret handshake involves pairs \( (x, y) \) that when squared and added, total 9.

To identify whether a pair belongs to this exclusive club, we simply test the handshake—or in mathematical terms, plug the values into our condition \( x^2 + y^2 = 9 \). All pairs that can perform the handshake perfectly (satisfy the condition) are admitted as members of the relation. In the textbook problem, the integer pair set \( \{(0,3),(0,-3),(3,0),(-3,0)\} \) was successfully tested providing evidence that these pairs are indeed part of the integer set relation defined by the specified condition.

Evaluating Mathematical Relations

Evaluating mathematical relations is akin to being a detective, where each pair of numbers is a clue that must align with the mathematical rule of the relation. In our textbook problem, the detective work involves checking if given pairs make the equation \( x^2 + y^2 = 9 \) come true.

To crack the case, you plug in the 'x' and 'y' from each pair into the equation and see if they fit. If they do, the pair is in conformity with the relation. It's much like following a recipe—only the ingredients that yield the desired result belong in the dish. The steps provided in the solution systematically evaluate each pair, like a detective verifying alibis, to ensure that they align with our mathematical 'secret handshake' from the integer set relations. This step-by-step approach ensures no stone is left unturned (or in our case, no integer pair is left untested) and leads to a clear verdict on which statements are true and which are not.