Mathematical Foundations: A Compact Glossary

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Functions, injections, surjections, and bijections

InjectiveSurjectiveBijective

A function from A to B

Choose a property to see how the arrows cover A and B.

f: A → B

This is a compact glossary of 19 ideas that recur across mathematics—not a complete textbook or a sequence of analysis theorems. The examples show how the definitions fit together; later courses supply their broader context.

Sets and set operations

1. Set

A set is a collection of distinct objects, called its elements. The notation $x\in X$ says that $x$ belongs to $X$. For example, $2\in\{1,2,3\}$.

2. Extensional and intensional definitions

A set may be described by listing its elements (extensionally) or by giving a property that characterizes them (intensionally). For instance, the even integers from 2 through 8 can be written as $\{2,4,6,8\}$ or $\{n\in\mathbb Z:2\le n\le8,\ n\text{ is even}\}$.

3. Subset

$X$ is a subset of $Y$, written $X\subseteq Y$, if every element of $X$ belongs to $Y$:

$X\subseteq Y\iff \forall x\,(x\in X\Rightarrow x\in Y).$

In particular, $\{1,2\}\subseteq\{1,2,3\}$.

4–7. Union, intersection, difference, and complement

The union collects elements in at least one set, while the intersection collects elements common to both:

$X\cup Y=\{x:x\in X\text{ or }x\in Y\},\qquad X\cap Y=\{x:x\in X\text{ and }x\in Y\}.$

The difference $X\setminus Y$ contains elements of $X$ not in $Y$. A complement needs a fixed universe $U$; for $X\subseteq U$, it is $X^c=U\setminus X$. For $X=\{1,2\}$, $Y=\{2,3\}$, the difference $X\setminus Y=\{1\}$ and intersection is $\{2\}$.

8. Empty set

The empty set $\emptyset$ has no elements and is a subset of every set.

9. Power set

The power set $\mathcal P(X)=\{A:A\subseteq X\}$ is the set of all subsets of $X$ (also written $2^X$). If $X$ is finite, then $|\mathcal P(X)|=2^{|X|}$. For example, $\mathcal P(\{a\})=\{\emptyset,\{a\}\}$.

10. Cartesian product

The Cartesian product consists of ordered pairs:

$A\times B=\{(a,b):a\in A,\ b\in B\}.$

More generally, $X_1\times\cdots\times X_n$ contains tuples with $i$th entry in $X_i$; if all factors are $X$, write $X^n$. Order matters: $(a,b)$ and $(b,a)$ need not be equal.

11. Symmetric difference

The symmetric difference contains elements in exactly one of $A,B$:

$A\triangle B=(A\setminus B)\cup(B\setminus A)=(A\cup B)\setminus(A\cap B).$

For the two sets above, $X\triangle Y=\{1,3\}$.

Algebraic structures and numbers

12. Groups and order

A group is a nonempty set $G$ with a binary operation $\cdot:G\times G\to G$ that is associative, has an identity $e$, and gives each element an inverse. Thus $(xy)z=x(yz)$, $ex=xe=x$, and $xx^{-1}=x^{-1}x=e$. A semigroup has an associative operation; a monoid is a semigroup with an identity. A group is abelian when the operation is commutative. The order of a finite group is its number of elements. The integers under addition form an abelian group with identity $0$.

13. Algebraic and transcendental numbers

A complex number is algebraic if it is a root of a nonzero polynomial with rational coefficients; otherwise it is transcendental. For example, $\sqrt2$ is algebraic because it is a root of $x^2-2$.

Functions and size

14. Function

A function $f:X\to Y$ assigns exactly one element of $Y$ to each $x\in X$. The sets $X,Y$ are its domain and codomain, and its image is $f(X)=\{f(x):x\in X\}\subseteq Y$. For $f(x)=x^2$ on $\mathbb R$, the image is $[0,\infty)$, not all of its codomain $\mathbb R$.

15. Injective, surjective, and bijective functions

A function is injective if $f(x_1)=f(x_2)$ implies $x_1=x_2$; surjective if every codomain element is an output; and bijective if both. The map $f:\mathbb R\to\mathbb R$, $f(x)=x^3$, is bijective.

16. Inverse function

A function has an inverse $f^{-1}:Y\to X$ exactly when it is bijective. The identities are $f^{-1}(f(x))=x$ and $f(f^{-1}(y))=y$. For example, the inverse of $x\mapsto x^3$ is $y\mapsto\sqrt[3]{y}$.

17. Equinumerous sets

Sets $A,B$ are equinumerous, written $A\sim B$, when a bijection pairs their elements. For finite sets this means they have the same number of elements.

18. Cardinality and countability

The cardinality $|A|$ is the size of $A$; for a finite set it is its number of elements. An infinite set in bijection with $\mathbb N$ is countably infinite. “Countable” commonly includes finite sets as well. The even integers are countably infinite: they can be listed as $0,2,-2,4,-4,\ldots$, giving a bijection with $\mathbb N$ after choosing its starting convention.

19. Comparing cardinalities

An injection $A\to B$ implies $|A|\le|B|$; equal cardinalities mean there is a bijection. In the usual cardinal arithmetic, $|A|<|B|$ means $|A|\le|B|$ and $|A|\ne|B|$. These definitions are a starting vocabulary for comparing sets, not a treatment of infinite cardinal arithmetic.