Beginning Activity 2: The Cartesian Product of Two Sets

Beginning Activity Beginning Activity 2: The Cartesian Product of Two Sets
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In Beginning Activity 1, we worked with ordered pairs without providing a formal definition of an ordered pair. We instead relied on your previous work with ordered pairs, primarily from graphing equations with two variables. Following is a formal definition of an ordered pair.

Definition.

Let $A$ and $B$ be sets. An ordered pair (with first element from $A$ and second element from $B$ ) is a single pair of objects, denoted by $(a, b)$ , with $a \in A$ and $b \in B$ and an implied order. This means that for two ordered pairs to be equal, they must contain exactly the same objects in the same order. That is, if $a, c \in A$ and $b, d \in B,$ then

(a, b) = (c, d) if and only if a = c and b = d .

The objects in the ordered pair are called the coordinates of the ordered pair. In the ordered pair $(a, b),$ $a$ is the first coordinate and $b$ is the second coordinate.

We will now introduce a new set operation that gives a way of combining elements from two given sets to form ordered pairs. The basic idea is that we will create a set of ordered pairs.

Definition.

If $A$ and $B$ are sets, then the Cartesian product, $A \times B,$ of $A$ and $B$ is the set of all ordered pairs $(x, y)$ where $x \in A$ and $y \in B .$ We use the notation $A \times B$ for the Cartesian product of $A$ and $B,$ and using set builder notation, we can write $A \times B = {(x, y) ∣ x \in A and y \in B} .$ We frequently read $A \times B$ as “ $A$ cross $B .$ ” In the case where the two sets are the same, we will write $A^{2}$ for $A \times A .$ That is,

A^{2} = A \times A = {(a, b) ∣ a \in A and b \in A} .

Let $A = {1, 2, 3}$ and $B = {a, b} .$

1.

Is the ordered pair $(3, a)$ in the Cartesian product $A \times B ?$ Explain.

2.

Is the ordered pair $(3, a)$ in the Cartesian product $A \times A ?$ Explain.

3.

Is the ordered pair $(3, 1)$ in the Cartesian product $A \times A ?$ Explain.

4.

Use the roster method to specify all the elements of $A \times B .$ (Remember that the elements of $A \times B$ will be ordered pairs.)

5.

Use the roster method to specify all of the elements of the set $A \times A = A^{2} .$

6.

For any sets $C$ and $D,$ explain carefully what it means to say that the ordered pair $(x, y)$ is not in the Cartesian product $C \times D .$

Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

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Beginning Activity Beginning Activity 2: The Cartesian Product of Two Sets
A4 US

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Beginning Activity Beginning Activity 2: The Cartesian Product of Two Sets A4US

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Beginning Activity Beginning Activity 2: The Cartesian Product of Two Sets
A4 US