Analyzing Function Properties: One-one, Many-one, Onto, and Into

Let's examine the given function \(f: \mathbb{R} \to \{0, 1\}\). The function is defined as:

\[\mathrm{F}(\mathrm{x})=\left\{\begin{array}{c} 1, \text {if x rational number } \\ 0, \text { irrational number } \end{array}\right.\]

We need to determine if this function possesses specific function properties, namely if it is one-one or many-one, and if it is onto or into. Understanding these function properties is crucial in mathematics.

Determining if the Function is One-one or Many-one

A function is considered one-one (or injective) if distinct elements in the domain map to distinct elements in the codomain. In other words, if \(f(x_1) = f(x_2)\), then \(x_1\) must be equal to \(x_2\).

A function is considered many-one if two or more distinct elements in the domain map to the same element in the codomain. This is the opposite of being one-one.

Let's look at the function \(f(x)\):

• For any rational number \(x\), \(f(x) = 1\). Examples of rational numbers include 1, 2, 1/2, -3/4. So, \(f(1) = 1\), \(f(2) = 1\), \(f(1/2) = 1\). Here, different inputs (1, 2, 1/2) give the same output (1).
• For any irrational number \(x\), \(f(x) = 0\). Examples of irrational numbers include \(\sqrt{2}\), \(\pi\), \(e\). So, \(f(\sqrt{2}) = 0\), \(f(\pi) = 0\), \(f(e) = 0\). Here, different inputs (\(\sqrt{2}\), \(\pi\), \(e\)) give the same output (0).

Since multiple distinct rational numbers map to 1 and multiple distinct irrational numbers map to 0, the function is not one-one. Therefore, the function is many-one. This is a key aspect of the function properties.

Determining if the Function is Onto or Into

A function is considered onto (or surjective) if every element in the codomain has at least one pre-image in the domain. In other words, for every element \(y\) in the codomain, there exists at least one element \(x\) in the domain such that \(f(x) = y\).

A function is considered into if there is at least one element in the codomain that does not have a pre-image in the domain. This is the opposite of being onto.

The codomain of the function \(f(x)\) is given as \(\{0, 1\}\).

• Does the element 0 in the codomain have a pre-image in the domain \(\mathbb{R}\)? Yes, any irrational number \(x\) serves as a pre-image because \(f(x) = 0\) for all irrational numbers. Since irrational numbers exist in \(\mathbb{R}\), 0 has pre-images.
• Does the element 1 in the codomain have a pre-image in the domain \(\mathbb{R}\)? Yes, any rational number \(x\) serves as a pre-image because \(f(x) = 1\) for all rational numbers. Since rational numbers exist in \(\mathbb{R}\), 1 has pre-images.

Since every element in the codomain \(\{0, 1\}\) has at least one pre-image in the domain \(\mathbb{R}\), the function is onto. This analysis confirms another aspect of the function properties.

Conclusion on Function Properties

Based on our analysis of the function properties:

• The function is many-one because different inputs map to the same output.
• The function is onto because every element in the codomain \(\{0, 1\}\) has a pre-image in the domain \(\mathbb{R}\).

Therefore, the function is many-one and onto. Understanding these function properties helps classify the behavior of the function \(f(x)\).