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Summarize

Natural Integer Rational Real Complex Irrational: real numbers outside the rational subset, defined by non- periodic decimal expansion (√2, π, e).

A diagram showing the inclusion hierarchy of the numerical sets \( \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \) and the position of the irrational numbers \( \mathbb{I} \).

Types of Numbers

lower bound upper bound a₁ M A a₂ a₃ m

Upper and lower bounds of a subset \(A \subseteq \mathbb{R}\), with \(m\) representing a lower bound and \(M\) an upper bound. The elements of \(A\) lie between the two bounds on the real line, providing a geometric interpretation of bounded sets, supremum, and infimum.

Supremum and Infimum

For the open interval A = (0, 1) the supremum and infimum exist but are not attained. No element of A equals 0 or 1, so the maximum and minimum do not exist. For the closed interval B = [0, 1] the endpoints belong to the set, so the infimum and supremum are attained and coincide with the minimum and maximum. inf A Sup A A = (0,1) inf A = min A Sup A = max A B = [0,1]

Comparison between supremum and maximum, and between infimum and minimum, using the intervals \((0,1)\) and \([0,1]\).

Supremum and Infimum

The function | x | is symmetric with respect to the y-axis. This symmetry implies that the function is even. It has an absolute minimum at the origin, but it is not differentiable there. x y=|x| y

The graph of \(y = |x|\), formed by two half-lines meeting at the origin and symmetric with respect to the \(y\)-axis.

Absolute Value

0 1 2 3 -1 -2 -3 1/3 π Rational Natural Integer −√3 Irrational Irrational Integers extend the natural numbers by including negative counterparts, forming a symmetric set around zero.

The integers as evenly spaced points on the number line, extending the natural numbers symmetrically about zero.

Integers

Geometrically, the absolute value of x, written |x|, represents the distance between x and 0 on the number line. 0 1 2 3 -1 -2 -3

The real line with its integer marks, where \(|x|\) is the distance between \(x\) and \(0\).

Real Numbers

A B U A ∪ B

Venn diagram of the union \(A \cup B\) inside a universal set \(U\).

Sets

A B U A ∩ B

Venn diagram of the intersection \(A \cap B\).

Sets

A U Aᶜ

Venn diagram of the complement \(A^c\) with respect to the universal set \(U\).

Sets

A B U A \ B

Venn diagram of the difference \(A \setminus B\).

Sets

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