Given two sets $A$ and $B$, the Cartesian Product of $A$ and $B$ denoted as $A \times B$ is the set of pairs $(a,b)$ such that $a \in A$ and $b \in B.$ For the cartesian product of a few sets like $A \times B \times C \times D$ $= \{(a,b,c,d): a \in A,$ $b \in B$ $c \in C$ $d \in D\}.$ If you have a long or countably infinite list of sets indexed in order like $S_1, S_2, ...,$ then the cartesian product of them can be written like $\prod\limits_{i = 1}^{n}S_i$ or $\prod\limits_{i = 1}^{\infty}S_i.$ Note that these would produce sets of sequences with $n$ terms or infinitely long sequences respectively, and the order of the terms matters. In essence, the first term in the sequence must be in $S_1$ and so on such that the $i$th term must be in $S_i.$
Problem 1: Let $A = \{1,2\}$ and $B = \{\pi, 2, 3\}.$ List all the elements of $A \times B.$ Solution
$A \times B$ $= \{(1,\pi),$ $(1,2),$ $(1,3),$ $(2,\pi),$ $(2,2),$ $(2,3)\}$
Problem 2: Let $A = \{1,2\}$ and $B = \{3,4\}$ and $C = \{5,6\}.$ List all the elements of $A \times B \times C.$ Solution
$A \times B \times C$ $=\{(1,3,5),$ $(1,3,6),$ $(1,4,5),$ $(1,4,6),$ $(2,3,5),$ $(2,3,6),$ $(2,4,5),$ $(2,4,6)\}$
Problem 3: If $A$ has $a$ elements in it, and $B$ has $b$ elements in it, how many elements are in $A \times B?$ Solution
$|A \times B|$ $= ab.$ In other words, there are $ab$ elements in $A \times B.$
Problem 4: If $S_i$ has $a_i$ elements in it, how many elements are in $\prod\limits_{i = 1}^{n}S_i?$ Solution
$\prod\limits_{i = 1}^{n}S_i$ has $\prod\limits_{i = 1}^{n}a_i$ $= a_1a_2...a_n$ elements in it.
Problem 5: If $A_1 \subseteq A_2$ and $B_1 \subseteq B_2$ and $C_1 \subseteq C_2,$ is $A_1 \times B_1 \times C_1$ a subset of $A_2 \times B_2 \times C_2?$ Solution
Yes.
Problem 6: Does $A \times B$ $=B \times A?$ In other words, is the cartesian product commutative? If not, give a counter example. Solution
No. $A = \{1\},$ $B = \{2\}.$ $(1,2) \neq (2,1)$
Problem 7: Does $(A \times B) \times C$ $=A \times (B \times C)?$ Solution
No. Elements in $(A \times B) \times C$ would be of the form $((a,b),c),$ whereas elements of $A \times (B \times C)$ would be of the form $(a,(b,c)).$
Problem 8: Is $\varnothing \times A$ $=\varnothing?$ Solution
Yes.
Problem 9: Does $A \times (B \cap C)$ $= (A \times B) \cap (A \times C)?$ What if you replaced $\cap$ with another set operation like $\cup$ or $\setminus?$ Solution
For those set operations, yes.
For a given set $S,$ $S^n$ is defined as $S \times S .... \times S$ or $\prod\limits_{i = 1}^{n}S.$ It can be thought of as the set of $n$-length sequences where each term is in $S.$ $S^\infty$ is defined as $\prod\limits_{i = 1}^{\infty}S.$ It can be thought of as the set of infinitely long sequences where each term is in $S.$
Problem 10: List all elements of the set $\{1,2\}^2.$ Solution
$\{1,2\}^2$ $=\{(1,1),$ $(1,2),$ $(2,1),$ $(2,2)\}$
Problem 11: If $A \subseteq B$, is $A^n$ a subset of $B^n?$ Solution
Yes.
Problem 12: Find a set $A$ such that $A^n = A.$ Solution
$A = \varnothing$
Problem 13: If $|A| = k,$ what is the cardinality of $A^n?$ In other words, how many elements are in $A^n?$ Solution
$|A^n|$ $=|A|^n$ $= k^n$
Problem 14: Find a non-empty set $A$ such that $|A^\infty|$ is finite. Solution
$A = \{a\},$ $|A^\infty|$ $=|\{(a,a,a,...)\}|$ $= 1$
Problem 15: Simplify $\varnothing^\infty$ and $\varnothing^n.$ Solution
$\varnothing^\infty$ $= \varnothing,$ $\varnothing^n$ $= \varnothing$