Question

Find the union \(C_{1} \cup C_{2}\) and the intersection \(C_{1} \cap C_{2}\) of the two sets \(C_{1}\) and \(C_{2}\), where: (a) \(C_{1}=\\{0,1,2,\\}, C_{2}=\\{2,3,4\),\(} .\) (b) \(C_{1}=\\{x: 0

Short answer

The solutions are: for part (a), the union is \{0,1,2,3,4\} and the intersection is \{2\}; for part (b), the union is the set of \(x\) satisfying \(0<x<3\) whereas the intersection includes the \(x\) that satisfy \(1 \leq x <2\); for part (c), for union \(C_{1} \cup C_{2}\) consists all ordered pairs within the borders defined by 0<x<3,1<y<3 and the intersection \(C_{1} \cap C_{2}\) includes ordered pairs that satisfy 1 < x < 2 and 1 < y < 2.

Step-by-step solution

Evaluate the Union and Intersection for Part (a)

For sets with distinct elements as in C1 and C2, the union \(C_{1} \cup C_{2}\) is the set containing all elements from both sets. In this case, this would be \{0,1,2,3,4\}. The intersection \(C_{1} \cap C_{2}\), on the other hand, contains only the elements that both sets share. For part a, this is \{2\}.

Evaluate the Union and Intersection for Part (b)

These sets are represented as intervals on the real number line. Hence, the union of \(C_{1}\) and \(C_{2}\) for part b is \(0<x<3\). For the intersection, it includes the overlapping interval between the two, which is \(1 \leq x <2\).

Evaluate the Union and Intersection for Part (c)

In this step, individual elements are ordered pairs instead of numbers. For part c, \(C_{1} \cup C_{2}\) includes all ordered pairs within the borders defined by 0<x<3 and 1<y<3. The intersection \(C_{1} \cap C_{2}\) includes only the pairs that lie within the overlapping regions, which amounts to 1<x<2 and 1<y<3. Hence, \(C_{1} \cap C_{2}\) includes ordered pairs that satisfy 1 < x < 2 and 1 < y < 2.

Key concepts

Union of Sets

When we talk about the union of sets, we are referring to a combination of all elements from two sets. The union is represented by the symbol \( \cup \). Imagine two circles representing sets with some overlap; the union includes everything inside both circles. This means every element from each set, without duplicates.
To apply this to our example, we have two parts:

• For part (a), sets \( C_{1} \) and \( C_{2} \) are \( \{0,1,2\} \) and \( \{2,3,4\} \) respectively. The union \( C_{1} \cup C_{2} \) is simply \( \{0,1,2,3,4\} \). We combine all elements.
• For part (b), \( C_{1} \) involves numbers between 0 and 2, whereas \( C_{2} \) contains values from 1 to less than 3. The union is expressed as \( 0 < x < 3 \), encompassing the entire stretch of numbers without gaps.
• For part (c), sets involve ordered pairs like coordinates. Combining all valid coordinates from both conditions yields a larger rectangle in the plane, from \( 0

This broad combination in each context illustrates the idea that the union takes everything once, encompassing all elements or points related to the defined parameters.

Intersection of Sets

The intersection of sets finds what they share. This concept is depicted using the symbol \( \cap \). In simpler words, it represents only the common elements between the involved sets.

In our exercise, let's break it down:

• For part (a), consider \( C_{1} = \{0,1,2\} \) and \( C_{2} = \{2,3,4\} \). Both sets include the number 2, so their intersection \( C_{1} \cap C_{2} \) is \( \{2\} \).
• In part (b), these sets represent intervals. \( C_{1} \) spans numbers slightly above 0 up to less than 2, and \( C_{2} \) starts at 1 and up to less than 3. The overlap, which is the intersection, lies between \( 1 \leq x < 2 \).
• For part (c), the situation includes x and y coordinate pairs. The shared area defined by individual inequalities \( 1

Each case presents a unique setup, but they all prioritize the shared component, whether numbers on a line or points in a space.

Ordered Pairs

Order of elements matters for ordered pairs. Unlike simple numbers, the order in which elements appear in an ordered pair \( (x, y) \) is crucial. The first element typically represents the x-coordinate, while the second represents the y-coordinate in a 2D space. This is important when visualizing or solving exercises involving these pairs.

The importance of this can be seen in part (c) of our exercise:

• Both sets \( C_{1} \) and \( C_{2} \) are defined by conditions on x and y that need to be satisfied simultaneously. For \( C_{1} \), this implies points where \( 0 < x < 2 \) and \( 1 < y < 2 \).
• In \( C_{2} \), the domain changes to \( 1 < x < 3 \) and \( 1 < y < 3 \).
• Understanding these conditions helps determine the union and intersection areas covered by these pairs.

Ordered pairs thus offer a structured way to define locations in a plane. They help in organizing information for problems involving spatial understanding or graph-based solutions.