Question

Find the elements in each of the following relations defined on \(\mathbb{R}\) : (a) \((x, y) \in R\) if and only if \(x+1

Short answer

(a) Pairs \((x, y)\) where \(y > x+1\). (b) Pairs \((x, y)\) with \(y < 0\) or \(x \leq \frac{3}{2}\). (c) Triplets \((x, y, z)\) with \(z = x^2 + y\).

Step-by-step solution

Understand the Relation for Part (a)

The relation given is \((x, y) \in R\) if and only if \(x + 1 < y\). This relation means that for any pair \((x, y)\), \(y\) must be greater than \(x + 1\). So, the elements \((x, y)\) in the relation are those where \(y\) is strictly greater than \(x + 1\).

Understand the Relation for Part (b)

The relation here is \((x, y) \in R\) if and only if \(y < 0\) or \(2x \leq 3\). This means that for \((x, y)\) to be in the relation, either \(y\) must be negative, or \(x\) must satisfy the inequality \(2x \leq 3\), which simplifies to \(x \leq \frac{3}{2}\). Either one of these conditions being true is sufficient for \((x, y)\) to be in the relation.

Understand the Relation for Part (c)

In this part, the relation is \((x, y, z) \in R\) if and only if \(x^2 + y = z\). This means that any triplet \((x, y, z)\) belongs to the relation if the sum of the square of \(x\) and \(y\) equals \(z\). Thus, \(z\) is always the sum of \(x^2\) and \(y\) for a triplet to be in the relation.

Key concepts

Real Numbers

Real numbers, denoted as \( \mathbb{R} \), are an essential part of mathematics. They include all the numbers on the number line. This set encompasses:

• Whole numbers (e.g., 0, 1, 2)
• Rational numbers (such as fractions \( \frac{1}{2} \), \( -\frac{3}{4} \))
• Irrational numbers (like \( \sqrt{2} \), \( \pi \), which cannot be expressed as simple fractions)
• Positive and negative numbers, including zero

Real numbers are used in equations and inequalities to represent continuous values across the number line. They offer a versatile palette for defining mathematical relations like those found in the exercise. In essence, whenever you see \( \mathbb{R} \), think of the most extensive kind of numbers we use without including imaginary components, providing a bridge from simple arithmetic to advanced calculus.

Inequalities

An inequality is a mathematical way to express the relationship between two values that are not equal. Instead of equalizing them, inequalities indicate that one value is either greater or smaller than the other. There are several types of inequalities:

• \( > \): Greater than
• \( < \): Less than
• \( \geq \): Greater than or equal to
• \( \leq \): Less than or equal to

For instance, in part (a) of the exercise, the inequality \( x + 1 < y \) implies that \( y \) is strictly greater than \( x + 1 \). In mathematics, inequalities help in defining the boundaries or limits of solutions and give shape to sets of solutions, which is vital in arranging values systematically. They are also pivotal in ordering real numbers, providing a comprehensive structure to the relationships of numbers.

Ordered Pairs

Ordered pairs are foundational elements in mathematics, especially when dealing with relations and functions. An ordered pair \( (x, y) \) contains two components, \( x \) and \( y \), typically representing a point in a coordinate plane. Key characteristics include:

• The first element \( x \) is commonly referred to as the abscissa.
• The second element \( y \) is called the ordinate.
• The order matters; \( (x, y) \) is not necessarily equal to \( (y, x) \).

In mathematical relations, ordered pairs define whether certain criteria are satisfied, as seen in parts (a) and (b) of the exercise. Understanding ordered pairs is essential for graphing functions, exploring geometric shapes, and establishing connections between sets.

Triplets

Triplets in mathematics extend the concept of ordered pairs by adding a third element. An ordered triplet \( (x, y, z) \) consists of three components: \( x \), \( y \), and \( z \). These are used to describe points in three-dimensional space. Key aspects of triplets include:

• The addition of the third dimension \( z \) allows for more complex relations compared to ordered pairs.
• Similar to ordered pairs, the sequence is crucial; changing the order changes the identity of the triplet.
• Triplets are essential in fields like physics where three dimensions are often involved.

In part (c) of the exercise, triplets are used where \( x^2 + y = z \). Here, triplets describe relationships where the third component is a result of a simple equation involving the first two components. Understanding triplets helps in visualizing and solving equations involving three variables.