Question

Given the intervals \(\mathrm{A}=\\{\mathrm{a} \mid 0 \leq \mathrm{a} \leq 4\\}, \mathrm{B}=\\{\mathrm{b} \mid 1 \leq \mathrm{b} \leq 2\\}\), and \(\mathrm{C}=\\{\mathrm{c} \mid-1<\mathrm{c}<2\\}\) find \(\mathrm{A} \times \mathrm{B}\) and \(\mathrm{B} \times \mathrm{C}\). Also graph \(\mathrm{A} \times \mathrm{C}\) and \((-\infty, 0] \times[0, \infty)\).

Short answer

The Cartesian products are: A × B = {(a, b) | 0 ≤ a ≤ 4, 1 ≤ b ≤ 2} B × C = {(b, c) | 1 ≤ b ≤ 2, -1 < c < 2} The graphical representation of A × C is a rectangle with vertices (0, -1), (4, -1), (4, 2), and (0, 2). The graphical representation of (-∞, 0] × [0, ∞) covers the area of quadrant II and the x and y-axes, extending infinitely.

Step-by-step solution

Understanding Intervals and Cartesian Products

Intervals are sets of real numbers between two endpoints. For example, the interval A contains all real numbers a such that \(0 \leq a \leq 4\). The Cartesian product of two sets A and B, denoted as A × B, contains all possible ordered pairs (a, b), where a is an element in A and b is an element in B.

Calculate A × B and B × C

To find the Cartesian products A × B and B × C, we identify the intervals' minimum and maximum values and note that any ordered pair with a and b from A and B respectively will form a rectangle. Similarly, for B × C, we'll identify b and c from their respective intervals. A × B: A = {a | 0 ≤ a ≤ 4} B = {b | 1 ≤ b ≤ 2} Thus, A × B = {(a, b) | 0 ≤ a ≤ 4, 1 ≤ b ≤ 2} B × C: B = {b | 1 ≤ b ≤ 2} C = {c | -1 < c < 2} So, B × C = {(b, c) | 1 ≤ b ≤ 2, -1 < c < 2}

Find A × C and (-∞, 0] × [0, ∞) Intervals

We have already found A × B and B × C. Now we need to define A × C and (-∞, 0] × [0, ∞) intervals. A × C: A = {a | 0 ≤ a ≤ 4} C = {c | -1 < c < 2} Thus, A × C = {(a, c) | 0 ≤ a ≤ 4, -1 < c < 2} (-∞, 0] × [0, ∞): Intuitively, this interval includes all ordered pairs where the first number is less than or equal to 0 and the second number is greater than or equal to 0. (-∞, 0] × [0, ∞) = {(x, y) | x ≤ 0, y ≥ 0}

Graph A × C and (-∞, 0] × [0, ∞)

We'll represent A × C and (-∞, 0] × [0, ∞) on the Cartesian plane. A × C = {(a, c) | 0 ≤ a ≤ 4, -1 < c < 2} On the Cartesian plane, draw a rectangle with vertices (0, -1), (4, -1), (4, 2) and (0, 2). This region represents all the ordered pairs in A × C. (-∞, 0] × [0, ∞) = {(x, y) | x ≤ 0, y ≥ 0} On the Cartesian plane, draw the area including the x-axis (x = 0, y ≥ 0) and y-axis (x ≤ 0, y = 0) and extending infinitely in their respective quadrants. This region represents all the ordered pairs in (-∞, 0] × [0, ∞).

Key concepts

Intervals

Intervals are essential when dealing with real numbers, as they help us define a range of values between two endpoints. In mathematics, an interval includes all numbers lying between these endpoints, which can be expressed in different ways depending on whether they include the boundary points or not. For example, when we describe the interval \(A = \{a \mid 0 \leq a \leq 4\}\), we're specifying all numbers from 0 to 4, inclusive. This type of interval is called a "closed" interval since it includes both endpoints.
On the other hand, interval \(C = \{c \mid -1 < c < 2\}\) is an "open" interval since it doesn't include its endpoints, meaning all numbers between -1 and 2 but not -1 or 2 themselves. By understanding these intervals, we can efficiently determine which numbers form part of a particular set.

Ordered Pairs

In mathematics, an ordered pair is simply a pair of elements written in a specific sequence, typically as \((a, b)\). This order is important because \((a, b)\) is different from \((b, a)\) unless both elements are equal. Ordered pairs are crucial in defining Cartesian products, which are sets composed of these pairs.
For example, when discussing the Cartesian product \(A \times B\), we're combining each element \(a\) from set \(A\) with each element \(b\) from set \(B\). The result is a new set of ordered pairs such as \((a, b)\), deriving from sets defined by their intervals.
This means for \(A \times B\) where \(A = \{a \mid 0 \leq a \leq 4\}\) and \(B = \{b \mid 1 \leq b \leq 2\}\), we create pairs like \((0,1)\), \((0,2)\), \((1,1)\), continuing through all combinations within those defined intervals. These pairs can form geometric shapes, such as rectangles, when plotted on a graph.

Graphing

Graphing is a visual way of representing mathematical concepts, making relationships and functions easier to understand. When graphing Cartesian products of intervals, we often use a coordinate plane. In this plane, each ordered pair \((x, y)\) is plotted as a point, depicting the spatial relation of the two elements.
Let's consider the graph of \(A \times C = \{(a, c) \mid 0 \leq a \leq 4, -1 < c < 2\}\). This involves drawing a rectangle on the Cartesian plane with corners at \((0, -1)\), \((4, -1)\), \((4, 2)\), and \((0, 2)\). This rectangle encapsulates all ordered pairs meeting the interval conditions.
Similarly, for \((-\infty, 0] \times [0, \infty)\), you highlight an entire quadrant of the plane stretching infinitely in the left-down direction. This visual tool helps us identify the range and domain of values in our sets and understand their relationships spatially.