Relations and Functions

February 18, 2026    admin

1. Relations

Let $A, B$ be sets. A relation $R$ from $A$ to $B$ is a subset of the Cartesian product:

$R \subseteq A \times B$

If $(a,b) \in R$, we write $aRb$.

Domain and Range

Domain: $\text{Dom}(R)=\{a\in A:\exists b\in B,\ (a,b)\in R\}$

Range: $\text{Ran}(R)=\{b\in B:\exists a\in A,\ (a,b)\in R\}$

Properties of Relations (on a set $A$)

• Reflexive: $\forall a\in A,\ aRa$
• Symmetric: $aRb \Rightarrow bRa$
• Antisymmetric: $aRb \text{ and } bRa \Rightarrow a=b$
• Transitive: $aRb \text{ and } bRc \Rightarrow aRc$

2. Order Relation

A relation $\le$ on $A$ is a partial order if it is:

• Reflexive
• Antisymmetric
• Transitive

A totally ordered set satisfies additionally:

$\forall a,b\in A,\ a\le b \text{ or } b\le a$

3. Equivalence Relations

A relation $\sim$ on $A$ is an equivalence relation if it is:

• Reflexive
• Symmetric
• Transitive

Equivalence Class

$[a]=\{x\in A: x\sim a\}$

The set of all equivalence classes forms a partition of $A$.

4. Functions

A function $f:A\to B$ is a relation such that:

$\forall a\in A,\ \exists! b\in B \text{ with } f(a)=b$

Image and Preimage

For $S\subseteq A$:

$f(S)=\{f(x):x\in S\}$

For $T\subseteq B$:

$f^{-1}(T)=\{x\in A: f(x)\in T\}$

5. Injective, Surjective, Bijective

• Injective (one-to-one): $f(x_1)=f(x_2)\Rightarrow x_1=x_2$
• Surjective (onto): $\forall y\in B,\ \exists x\in A,\ f(x)=y$
• Bijective: Injective and Surjective

6. Inverse Function

If $f:A\to B$ is bijective, then there exists:

$f^{-1}:B\to A$

such that:

$f^{-1}(f(x))=x,\quad f(f^{-1}(y))=y$

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Real Number System

1. Field Properties of $\mathbb{R}$

For all $a,b,c\in\mathbb{R}$:

• Closure under $+$ and $\cdot$
• Associativity: $(a+b)+c=a+(b+c)$
• Commutativity: $a+b=b+a$, $ab=ba$
• Distributive: $a(b+c)=ab+ac$
• Additive identity: $a+0=a$
• Multiplicative identity: $a\cdot1=a$
• Additive inverse: $a+(-a)=0$
• Multiplicative inverse: $a\ne0 \Rightarrow a\cdot a^{-1}=1$

2. Order Properties

• Trichotomy: Exactly one of $ab$
• If $a
• $a
• $a0 \Rightarrow ac

3. Natural Numbers

$\mathbb{N}=\{1,2,3,\dots\}$

Principle of Mathematical Induction:

• Base step
• If $P(n)\Rightarrow P(n+1)$, then $P(n)$ holds for all $n\in\mathbb{N}$

4. Integers and Rational Numbers

$\mathbb{Z}=\{\dots,-2,-1,0,1,2,\dots\}$

$\mathbb{Q}=\left\{\frac{p}{q}:p,q\in\mathbb{Z},\ q\ne0\right\}$

5. Absolute Value

$|x|= \begin{cases} x, & x\ge0\\ -x, & x<0 \end{cases}$

Properties

• $|x|\ge0$
• $|xy|=|x||y|$
• $|x+y|\le |x|+|y|$ (Triangle inequality)
• $||x|-|y||\le |x-y|$

6. Basic Inequalities

Inequality of Means

For $a,b>0$:

$\frac{a+b}{2}\ge\sqrt{ab}$

Power Inequality

If $0<a<b$ and $n\in\mathbb{N}$, then

$$a^n < b^n$$

Cauchy–Schwarz Inequality

For $x_i,y_i\in\mathbb{R}$:

$\left(\sum_{i=1}^n x_i y_i\right)^2 \le \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right)$

Chebyshev’s Inequality

If sequences are similarly ordered:

$\frac{1}{n}\sum_{i=1}^n a_i b_i \ge \left(\frac{1}{n}\sum_{i=1}^n a_i\right) \left(\frac{1}{n}\sum_{i=1}^n b_i\right)$

Weierstrass Inequality

For $x>-1$ and $n\in\mathbb{N}$:

$(1+x)^n \ge 1+nx$

Mathematics   Tagged   Relations & Functions