Nonlinear finite elements/Sets

A familiarity with the notation of sets is essential for the student who wants to read modern literature on finite elements. This handout gives you a brief review of set notation. More details can be found in books on advanced calculus.

Sets

A set is a well-defined collection of objects. As far as we are concerned, these objects are mainly numbers, vectors, or functions.

If an object $x$ is a member of a set $A$, we write

$x \in A ~~~~~~~~(x~\text{belongs to}~A)~.$

If $x$ is not a member of $A$, we write

$x \notin A ~~~~~~~~(x~\text{does not belong to}~A)~.$

An example of a finite set (of functions) is

$S = \{x, \sin(x), \cos(x), \exp(x), \log(x)\}~.$

Another example is the set of integers greater than 5 and less than 12

$B = \{6, 7, 8, 9, 10, 11\}~.$

If we denote the set of all integers by $\mathbb{Z}$, then we can alternatively write

$B = \{n ~|~ n \in \mathbb{Z}, 5 < n < 12\}~.$

The set $\mathbb{Z}^{+}$ of positive integers is an infinite set and is written as

$\mathbb{Z}^{+} = \{n ~|~ n \in \mathbb{Z}, n > 0\}~.$

An empty (or null) set is a set with no elements. It is denoted by $\varnothing$. An example is

$\varnothing = \{n ~|~ n \in \mathbb{Z}^{+}, n < 0\}~ = \{\}~.$

Subsets

If $A$ and $B$ are two sets, then we say that $A$ is a subset of $B$ if each element of $A$ is an element of $B$.

For example, if the two sets are

$A = \{\sin(x), \cos(x)\}~ B = \{x, \sin(x), \cos(x), \exp(x), \log(x)\}~~\text{and}~~$

we write

$A \subset B ~~~~~(A~\text{is a proper subset of}~B)~.$

On the other hand, if $A$ is a subset of $B$ which may be the set $B$ itself we write

$A \subseteq B ~~~~~(A~\text{is a subset of}~B)~.$

If $A$ is not a subset of $B$, we write

$A \not\subset B ~~~~~(A~\text{is not a subset of}~B)~.$

Equality of sets

Two sets $A$ and $B$ are equal if they contain exactly the same elements. Thus,

$A = B \iff A \subseteq B ~\text{and}~ B \subseteq A ~.$

The symbol $\iff$ means if and only if.

For example, if

$A = \{x ~|~ x^2 = 4\} ~\text{and}~ B = \{2, -2\}$

then $A = B$.

Union, Intersection, Difference of Sets

The union of two sets $A$ and $B$ is the set of all elements that are in $A$ or $B$.

$C = A \cup B = \{x ~|~ x \in A ~\text{or}~ x \in B\}~.$

The intersection of two sets $A$ and $B$ is the set of all elements that are both in $A$ and in $B$.

$D = A \cap B = \{x ~|~ x \in A ~\text{and}~ x \in B\}~.$

The difference of two sets $A$ and $B$ is the set of all elements that are in $A$ but not in $B$.

$E = A - B = \{x ~|~ x \in A ~\text{and}~ x \not\in B\}~.$

The complement of a set $A$ (denoted by $A^{'}$) is the set of all elements that are not in $A$ but belong to a larger universal set $U$.

$A^{'} = U - A = \{x ~|~ x \not\in A\}~.$

Countable Sets

Suppose we have a set $A$. Such a set is called countable if each of its members can be labeled with an integer subscript of the form

$A = \{a_1, a_2, a_3, a_4, \dots\}~.$

Obviously, each finite set is countable. Some infinite sets are also countable. For instance, the set of integers is countable because you can label each integer with an subscript that is also an integer. However, you cannot do that with the real numbers which are uncountable.

The set of functions

$P = \{f_k ~|~ f_k(x) = x^k, k=1,2,\dots\}$

is countable.

The set of points on the real line

$A = \{x ~|~ 0 \le x \le 1\}$

is not countable because the points cannot be labeled $a_1$, $a_2$, $\dots$.

Cartesian Product

The Cartesian product of two sets $A$ and $B$ is the set of all ordered pairs $(a,b)$, such that

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

In general, $A \times B \neq B \times A$.

For example, if

$A = \{1, 5, 7\} ~\text{and}~ B = \{3, 6\}$

then

$A \times B = \{(1,3),(1,6),(5,3),(5,6),(7,3),(7,6)\}$

and

$B \times A = \{(3,1),(3,5),(3,7),(6,1),(6,5),(6,7)\} \neq A \times B~.$

The Set of Real Numbers ($\mathbb{R}$)

The set of real numbers ($\mathbb{R}^{}$) can be visualized as an infinitely long line with each real number being represented as a point on this line.

We usually deal with subsets of $\mathbb{R}^{}$, called intervals.

Let $a$ and $b$ be two points on $\mathbb{R}^{}$ such that $a \le b$. Then,

• The open interval $(a,b)$ is defined as

$(a,b) = \{x ~|~ x \in \mathbb{R}, a < x < b\}~.$

• The closed interval $[a,b]$ is defined as

$[a,b] = \{x ~|~ x \in \mathbb{R}, a \le x \le b\}~.$

• The half-open intervals $(a,b]$ and $[a,b)$ are defined as

$(a,b] = \{x ~|~ x \in \mathbb{R}, a < x \le b\} ~\text{and}~ [a,b) = \{x ~|~ x \in \mathbb{R}, a \le x < b\}~.$

Let $p \in \mathbb{R}^{}$ and $\epsilon > 0$. Then the neighborhood of $p$ is defined as the open interval

$\text{nbd}(p;\epsilon) := (p - \epsilon, p+\epsilon) = \{ x ~|~ p - \epsilon < x < p + \epsilon \}$

Let $X \subset \mathbb{R}^{}$. Then $p$ is an interior point of $X$ if we can find a nbd($p$) all of whose points belong to $X$.

If every point of $X$ is an interior point, then $X$ is called an open set. For example, the interval $(a,b)$ is an open set. So is the real line $\mathbb{R}^{}$.

A set $X \subset \mathbb{R}^{}$ is called closed if its complement $X^{'} = \mathbb{R}^{} - X$ is open.

The closure $\bar{X}$ of a set $X \subset \mathbb{R}^{}$ is the union of the set and its boundary points (a rigorous definition of closed sets can be made using the concept of points of accumulation).

Open and Closed Sets in $\mathbb{R}^n$

The concept of the real line can be extended to higher dimensions. In two dimensions, we have $\mathbb{R}^{2}$ which is defined as

$\mathbb{R}^{2} = \mathbb{R}^{}\times\mathbb{R}^{} = \{(x,y) ~|~ x,y \in \mathbb{R}^{}\}~.$

$\mathbb{R}^{2}$ can be thought of as a two-dimensional plane and each member of the set $\mathbf{x} = (x,y)$ represents a point on the plane.

In three dimensions, we have

$\mathbb{R}^{3} = \mathbb{R}^{}\times\mathbb{R}^{}\times\mathbb{R}^{} = \{(x,y,z) ~|~ x,y,z \in \mathbb{R}^{}\}~.$

In $n$ dimensions, the concept is extended to mean

$\mathbb{R}^{n} = \mathbb{R}^{}\times\mathbb{R}^{}\times\mathbb{R}^{}\dots\times\mathbb{R}^{} = \{(x_1,x_2,x_3,\dots,x_n) ~|~ x_1,x_2,x_3,\dots,x_n \in \mathbb{R}^{}\}~.$

In the case of sets in $\mathbb{R}^{n}$ the concept of distance in $\mathbb{R}^{}$ is extended so that

$\text{nbd}(\mathbf{p};\epsilon) := \{ \mathbf{x} ~|~ \mathbf{p} \in \mathbb{R}^{n}, |\mathbf{x} - \mathbf{p}| < \epsilon \}$

where

$|\mathbf{x} - \mathbf{p}| = \sqrt{(x_1 - p_1)^2 + (x_2 - p_2)^2 + \dots + (x_n - p_n)^2}~.$

The definition of interior point also follows from the definition in $\mathbb{R}^{}$. Thus if $\Omega \subset \mathbb{R}^{n}$, then $p \in \mathbb{R}^{n}$ is an interior point if we can always find a nbd$(p;\epsilon)$, all of whose points belong to $\Omega$. If every point on $\Omega$ is an interior point, then $\Omega$ is an open set. As in the real number line, a closed set is the complement of an open set. One way of creating a closed set is by taking an open set $\Omega$ and its boundary $\Gamma$. This particular closed set is called the closure $\bar\Omega$ of $\Omega$. A rigorous definition can once again be obtained using the concept of points of accumulation.