Theory

Sets, induction and elementary functions

The language of sets and the number sets from N to R, the principle of induction and the binomial theorem, then the elementary functions with their graphs, the starting toolkit for all of analysis.

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The language of sets
Operations on sets

The language of sets

Set, membership, inclusion

A set is a collection of objects, called elements. The notation $x \in A$ means that $x$ is an element of $A$ ; $x \in / A$ means that it is not. A set is given by listing its elements, as in ${1, 2, 3}$ , or by stating the property that identifies them: ${x \in E : P (x)}$ is the set of the elements of $E$ for which $P (x)$ is true. The empty set $\emptyset$ is the set with no elements.

$A \subseteq B$ ( $A$ is contained in $B$ , or $A$ is a subset of $B$ ) means that every element of $A$ also belongs to $B$ . Equality of sets is equivalent to double inclusion:

A = B ⟺ {A \subseteq B B \subseteq A

Double inclusion is the method used to prove that two sets are equal: the two inclusions are verified separately.

A \subseteq B

: every element of

A

also belongs to

B

Example

$A = {x \in R : x^{2} < 4}$ . We have $1 \in A$ because $1^{2} = 1 < 4$ , and $3 \in / A$ because $3^{2} = 9 \geq 4$ . Moreover ${0, 1} \subseteq A$ , because both elements satisfy the condition $x^{2} < 4$ .

Warning

The symbols $\in$ and $\subseteq$ have different meanings: $\in$ relates an element to a set, $\subseteq$ relates two sets. For $A = {1, 2}$ the correct statements are $1 \in A$ and ${1} \subseteq A$ ; the statements $1 \subseteq A$ and ${1} \in A$ are false, because ${1}$ is not one of the two elements of $A$ and $1$ is not a set.

Operations on sets

1. Union, intersection, difference, complement

For $A$ and $B$ subsets of a universal set $E$ :

A \cup B = {x : x \in A or x \in B}, A \cap B = {x : x \in A and x \in B}, A ∖ B = {x : x \in A and x \in / B}, A^{c} = E ∖ A .

The “or” of the union is not exclusive: the elements common to $A$ and $B$ belong to the union. If $A \cap B = \emptyset$ , the sets are called disjoint. The complement depends on the universal set $E$ ; for subsets of $R$ , unless stated otherwise, the universal set is $R$ . The operations translate the logical connectives, therefore the solution set of a system of inequalities is the intersection of the solution sets of the individual inequalities, and the solution set of an inequality solved by cases is the union of the solutions of the cases.

Disjoint sets:

A \cap B = \emptyset

De Morgan's law

(A \cup B)^{c} = A^{c} \cap B^{c}

: the elements outside the union are outside

A

and outside

B

De Morgan's law

(A \cap B)^{c} = A^{c} \cup B^{c}

: the elements outside the intersection are outside

A

or outside

B

Distributive property

A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

Distributive property

A \cup (B \cap C) = (A \cup B) \cap (A \cup C)

Example

$A = {1, 2, 3, 4}$ and $B = {3, 4, 5}$ , with universal set $E = {1, 2, 3, 4, 5, 6}$ :

A \cup B = {1, 2, 3, 4, 5}, A \cap B = {3, 4}, A ∖ B = {1, 2}, B ∖ A = {5}, A^{c} = {5, 6} .

A \cup B = {1, 2, 3, 4, 5}

A \cap B = {3, 4}

A ∖ B = {1, 2}

(A ∖ B) \cup (B ∖ A) = {1, 2, 5}

: the elements that lie in only one of the two sets form the symmetric difference

A △ B

A^{c} = E ∖ A = {5, 6}

2. Cartesian product

The Cartesian product of $A$ and $B$ is the set of ordered pairs

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

In pairs the order matters, so in general $A \times B \neq = B \times A$ : for $A = {1, 2}$ and $B = {0}$ we have $A \times B = {(1, 0), (2, 0)}$ and $B \times A = {(0, 1), (0, 2)}$ . The set $R^{2} = R \times R$ is the Cartesian plane: the graphs of functions are subsets of it, therefore the Cartesian product is the setting in which functions are studied.

The theory continues with 20 more sections (the number sets, intervals, upper bounds lower bounds maximum and minimum…), plus all worked examples, the solutions and the PDF.

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