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Abstract Algebra

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Moderate

Cartesian Product: definition and properties, example

ABSALG-4QLMVN

The Cartesian product of two sets $A$ and $B$, denoted $A\times B$, is the set of ordered pairs $(a,b)$, with $a\in A$ and $b\in B$. Therefore $|A\times B|=|A||B|$.

In set notation:

$A\times B=\{(a,b)\mid a\in A\;{\rm and}\;b\in B\}$

Let $A=\{3\}$, $B=\{3,5,7\}$, $C=\{2,3,4\}$, $D=\{2,4\}$

Which of the following is true?

A

$A\times C=C\times A$

B

$(A\times A)\times A=A\times (A\times A)$

C

$A\times \emptyset = A$, where $\emptyset$ is the empty set.

D

$(A\cup B)\times (C\cup D)=(A\times C)\cup (B\times D)$ is the set of $9$ elements given by

$\{(3,2),(3,3),(3,4),(5,2),(5,3),(5,4),(7,2),(7,3),(7,4)\}$

E

$(A\cap B)\times (C\cap D)=(A\times C)\cap (B\times D)=\{(3,2),(3,4)\}$