Notation

Discrete Sets in DMU

A set is a collection of unique elements. The elements can be numbers, tuples, vectors, or any other objects. Examples: \[\{1, 2, 4\}, \{A, B, C\}\]
Two sets are equal if they contain the exact same elements:\[\{1, 2, 4\} = \{1, 4, 2\} \text{, but } \{1, 2, 4\} \neq \{1, 2, 3\}\]
The symbol \(\in\) is read "in", and indicates that an object is in a set, i.e. \[1 \in \{1,2\} \text{, but } 3 \notin \{1, 2\}\]
The union operator, \(\cup\), is used to combine two sets so that the new one contains all of the elements of both sets: \[\{1, 2, 3\} \cup \{1, 2, 4\} = \{1, 2, 3, 4\}\]
The Cartesian product, \(\times\), creates a new set of tuples of all possible combinations of the elements of the argument sets: \[\{1, 2, 3\} \times \{A, B\} = \{(1, A), (2, A), (3, A), (1, B), (2, B), (3, B)\}\]
An exponent is used to denote multiple applications of the Cartesian product operator: \[A^3 = A \times A \times A\]
Example: The state space for the grid world used on the homework is\[S_\text{grid world} = \{1, ..., 10\}^2 \cup (-1, -1)\] (since \((-1, -1)\) is used as a terminal state).

Some sets contain an uncountably infinite number of elements. These are often constructed with the set of all real numbers, \(\mathbb{R}\), or intervals of real numbers. In DMU, we refer to these sets as "continuous". Examples:
- \([0, 1]\) is the set of all real numbers between 0 and 1 including 0 and 1.
- \((-5, -4)\) is the set of all real numbers between -5 and -4, not including -5 and -4. (Confusingly, this clashes with the notation for tuples. Use context clues!)
Cartesian products of \(\mathbb{R}\) or intervals can be used to construct vector spaces:
- \(\mathbb{R}^3\) is the space of all 3-dimensional real-valued vectors.
- \([0, 1] \times [2, 3]\) is a "box" with the bottom left corner at \((0,2)\) and top right corner at \((1,3)\).
We can also combine continuous sets with discrete ones:
- \((0, 1) \cup \{0, 1\} = [0, 1]\)
- \([1, 2] \times \{3, 4.5\}\) is the set of all 2-dimensional vectors where the first entry is between 1 and 2 (inclusive) and the second entry is 3 or 4.5. The vectors \((1.42,3)\) and \((2, 4.5)\) are in this set, but \((0.56, 3)\) and \((1.42, 4)\) are not.