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<!DOCTYPE html>
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    <title>Relations and Functions - In-depth Overview</title>
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<header>
    Relations and Functions - In-depth Overview
</header>

<section>
    <h1>1. Relations</h1>
    <p>A <strong>relation</strong> is a connection or correspondence between elements of two sets. Formally, a relation from a set <code>A</code> to a set <code>B</code> is a subset of the Cartesian product <code>A × B</code>. This means that a relation consists of ordered pairs <code>(a, b)</code>, where <code>a ∈ A</code> and <code>b ∈ B</code>.</p>

    <h2>Examples of Relations:</h2>
    <p>Set <code>A = {1, 2, 3}</code>, Set <code>B = {a, b, c}</code>: A relation <code>R</code> from <code>A</code> to <code>B</code> could be <code>R = {(1, a), (2, b), (3, c)}</code>, meaning that 1 relates to a, 2 relates to b, and 3 relates to c.</p>

    <h2>Types of Relations:</h2>
    <ul>
        <li><strong>Reflexive Relation:</strong> A relation <code>R</code> on a set <code>A</code> is reflexive if every element in <code>A</code> is related to itself. In other words, <code>∀a ∈ A, (a, a) ∈ R</code>.</li>
        <div class="example">
            Example: If <code>A = {1, 2, 3}</code>, a reflexive relation would include <code>(1, 1), (2, 2), (3, 3)</code>.
        </div>

        <li><strong>Symmetric Relation:</strong> A relation <code>R</code> on set <code>A</code> is symmetric if for all <code>a, b ∈ A</code>, whenever <code>(a, b) ∈ R</code>, then <code>(b, a) ∈ R</code>.</li>
        <div class="example">
            Example: If <code>R = {(1, 2), (2, 1)}</code>, it is symmetric.
        </div>

        <li><strong>Antisymmetric Relation:</strong> A relation <code>R</code> is antisymmetric if for all <code>a, b ∈ A</code>, whenever <code>(a, b) ∈ R</code> and <code>(b, a) ∈ R</code>, then <code>a = b</code>.</li>
        <div class="example">
            Example: <code>R = {(1, 2), (2, 3)}</code> is antisymmetric because there is no case where both <code>(a, b)</code> and <code>(b, a)</code> are present with <code>a ≠ b</code>.
        </div>

        <li><strong>Transitive Relation:</strong> A relation <code>R</code> on set <code>A</code> is transitive if whenever <code>(a, b) ∈ R</code> and <code>(b, c) ∈ R</code>, then <code>(a, c) ∈ R</code>.</li>
        <div class="example">
            Example: If <code>R = {(1, 2), (2, 3), (1, 3)}</code>, the relation is transitive.
        </div>
    </ul>
</section>

<section>
    <h1>2. Functions</h1>
    <p>A <strong>function</strong> is a special type of relation where every element of the domain (set <code>A</code>) is related to exactly one element of the codomain (set <code>B</code>). The function can be written as <code>f: A → B</code>, where <code>f(x) = y</code> means that the element <code>x</code> in set <code>A</code> is mapped to the element <code>y</code> in set <code>B</code>.</p>

    <h2>Key Characteristics of Functions:</h2>
    <ul>
        <li><strong>Domain:</strong> The set of all possible inputs (values from set <code>A</code>).</li>
        <li><strong>Range:</strong> The set of all possible outputs (values from set <code>B</code>).</li>
        <li><strong>Codomain:</strong> The set <code>B</code>, which contains all possible outputs of the function.</li>
    </ul>

    <h2>Types of Functions:</h2>
    <ul>
        <li><strong>One-to-One (Injective):</strong> A function <code>f</code> is one-to-one if different elements of the domain map to different elements of the codomain. In other words, if <code>f(a₁) = f(a₂)</code>, then <code>a₁ = a₂</code>.</li>
        <li><strong>Onto (Surjective):</strong> A function <code>f</code> is onto if every element of the codomain has at least one corresponding element in the domain. In other words, the range of the function is equal to its codomain.</li>
        <li><strong>One-to-One Correspondence (Bijective):</strong> A function is bijective if it is both one-to-one (injective) and onto (surjective). This means that there is a one-to-one correspondence between all elements of the domain and the codomain.</li>
    </ul>

    <h2>Notation:</h2>
    <p><code>f: A → B</code> indicates that <code>f</code> is a function from set <code>A</code> to set <code>B</code>.</p>
    <p><code>f(x) = y</code> indicates that <code>f</code> maps <code>x</code> to <code>y</code>.</p>

    <h2>Examples of Functions:</h2>
    <ul>
        <li><code>f(x) = 2x + 3</code> is a function from the set of real numbers to the set of real numbers.</li>
        <li><code>f(x) = x²</code> is a function from real numbers to non-negative real numbers.</li>
    </ul>
</section>

<section>
    <h1>3. Important Formulas and Theorems</h1>

    <div class="important">
        <h2>Function Composition:</h2>
        <p>If you have two functions <code>f</code> and <code>g</code>, the composition <code>(f ∘ g)(x) = f(g(x))</code>.</p>
        <div class="code-example">
            Example: If <code>f(x) = 2x</code> and <code>g(x) = x + 1</code>, then <code>(f ∘ g)(x) = 2(x + 1) = 2x + 2</code>.
        </div>
    </div>

    <div class="important">
        <h2>Inverse Function:</h2>
        <p>If <code>f</code> is a function from <code>A</code> to <code>B</code>, the inverse function <code>f⁻¹</code> is a function from <code>B</code> to <code>A</code> that satisfies:</p>
        <div class="math-box">
            <code>f⁻¹(f(x)) = x</code> and <code>f(f⁻¹(y)) = y</code>
        </div>
        <div class="code-example">
            Example: If <code>f(x) = 2x + 3</code>, the inverse function is <code>f⁻¹(x) = (x - 3) / 2</code>.
        </div>
    </div>

    <div class="important">
        <h2>Domain and Range of Functions:</h2>
        <ul>
            <li><strong>Domain:</strong> The set of all possible input values.</li>
            <li><strong>Range:</strong> The set of all possible output values.</li>
        </ul>
    </div>

    <div class="important">
        <h2>Function Behavior:</h2>
        <ul>
            <li><strong>Increasing Function:</strong> A function <code>f(x)</code> is increasing if <code>f(x₁) < f(x₂)</code> whenever <code>x₁ < x₂</code>.</li>
            <li><strong>Decreasing Function:</strong> A function <code>f(x)</code> is decreasing if <code>f(x₁) > f(x₂)</code> whenever <code>x₁ < x₂</code>.</li>
            <li><strong>Constant Function:</strong> A function <code>f(x)</code> is constant if <code>f(x₁) = f(x₂)</code> for all <code>x₁, x₂</code> in the domain.</li>
        </ul>
    </div>

    <div class="important">
        <h2>Slope of a Function (For Linear Functions):</h2>
        <p>The slope <code>m</code> of a linear function <code>f(x) = mx + b</code> is given by the change in <code>y</code> divided by the change in <code>x</code>:</p>
        <div class="formula">
            <code>m = Δy / Δx</code>
        </div>
    </div>
</section>

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