Sets A, B, C: Enumeration And Properties Explained

Hey guys! Let's dive into a fun math problem involving sets. We've got three sets here: A, B, and C. Our goal is to figure out what's in set C and then check some statements about these sets. It's like a mathematical treasure hunt, so let's get started!

Understanding the Sets A, B, and C

First off, let's break down what each set contains. This is crucial for understanding the problem and tackling it effectively. Make sure you really grasp what each set represents before moving on.

• Set A: A = {0, 3, 6}. This set simply contains three numbers: 0, 3, and 6. Nothing too complicated here!
• Set B: B = {1, 2}. This set contains two numbers: 1 and 2. Again, pretty straightforward.
• Set C: C = {x ∈ N | x = a^b, a ∈ A and b ∈ B}. Okay, this one's a bit more interesting. Let's unpack it. This set C contains elements 'x'. These elements 'x' are natural numbers (that's what the 'N' means). Each 'x' can be calculated as 'a' raised to the power of 'b' (a^b), where 'a' comes from set A and 'b' comes from set B. Essentially, we're going to take each number from set A and raise it to the power of each number in set B to create set C.

The Importance of Clear Set Definitions

Understanding set notation is absolutely key in mathematics. When dealing with sets, you'll often encounter notation like this, and being able to decipher it quickly will save you a lot of headaches. The notation x ∈ N means "x is an element of the set of natural numbers." The vertical bar | is often read as "such that," so the whole definition of set C can be read as "C is the set of all x in the natural numbers such that x equals a to the power of b, where a is in A and b is in B."

a) Representing Set C by Enumerating Its Elements

Now for the fun part! We need to actually figure out what numbers are in set C. Remember, we're taking each element from set A and raising it to the power of each element from set B. Let's do this systematically.

We'll consider all possible combinations of 'a' from set A and 'b' from set B:

• Case 1: a = 0
  • b = 1: 0^1 = 0
  • b = 2: 0^2 = 0
• Case 2: a = 3
  • b = 1: 3^1 = 3
  • b = 2: 3^2 = 9
• Case 3: a = 6
  • b = 1: 6^1 = 6
  • b = 2: 6^2 = 36

So, based on these calculations, the possible elements of set C are {0, 3, 9, 6, 36}. Remember that in set theory, we only list unique elements. Even though 0 appears twice, we only include it once in the set.

Therefore, C = {0, 3, 6, 9, 36}. This is our enumerated set C!

Why Enumerating Sets is Important

Enumerating the elements of a set makes it much easier to work with. Instead of dealing with abstract definitions, you have a concrete list of elements. This is especially helpful when you need to compare sets, perform operations on them, or, as we'll see in part (b), check if certain statements about them are true.

b) Completing the Table: Analyzing Statements About the Sets

Okay, we've conquered the first part. Now, let's move on to the table. We've got a bunch of statements about our sets, and we need to figure out if they're true or false. We will discuss and explain each statement step by step to make sure everything is clear.

Let's create a table like the one described in the problem and go through each statement:

Now, let's fill in the table, one statement at a time.

1. 0∉A (0 is not an element of A)

• True or False? False
• Explanation: Remember, set A = {0, 3, 6}. The symbol '∈' means "is an element of," and '∉' means "is not an element of." Since 0 is clearly listed as an element of A, the statement "0 is not an element of A" is false. It's a direct contradiction of what we know about set A.

2. 0∈B (0 is an element of B)

• True or False? False
• Explanation: Set B = {1, 2}. Does 0 appear in this set? Nope! Therefore, the statement "0 is an element of B" is false. 0 is simply not part of set B.

3. A≠B (A is not equal to B)

• True or False? True
• Explanation: For two sets to be equal, they must contain exactly the same elements. Set A = {0, 3, 6} and set B = {1, 2}. These sets have completely different elements. Therefore, A is definitely not equal to B, and the statement is true. This is a fundamental concept in set theory.

4. A⊂N (A is a subset of N)

• True or False? True
• Explanation: This one requires a bit more understanding. The symbol '⊂' means "is a subset of." A set A is a subset of a set N if every element in A is also an element in N. In this case, N represents the set of natural numbers. Natural numbers are positive whole numbers (1, 2, 3, ...), and sometimes 0 is included, depending on the definition being used. Since A = {0, 3, 6}, and all these numbers can be considered natural numbers, the statement "A is a subset of N" is true.

5. Card C=6 (The cardinality of C is 6)

• True or False? False
• Explanation: The term "cardinality" (often written as "Card") refers to the number of elements in a set. We found that C = {0, 3, 6, 9, 36}. How many elements are in C? There are 5 elements. Therefore, the statement "Card C = 6" is false. The cardinality of C is actually 5.

6. 36∈C (36 is an element of C)

• True or False? True
• Explanation: We found that C = {0, 3, 6, 9, 36}. Is 36 in this set? Yes! Therefore, the statement "36 is an element of C" is true. This is a simple check to see if we correctly calculated the elements of set C.

7. 8∈C (8 is an element of C)

• True or False? False
• Explanation: Again, we have C = {0, 3, 6, 9, 36}. Is 8 in this set? Nope! Therefore, the statement "8 is an element of C" is false. 8 is not one of the numbers we generated by raising elements of A to powers from B.

The Completed Table

Okay, we've analyzed each statement carefully. Let's put it all together in our completed table:

Double-Checking Your Work

In math, it's always a good idea to double-check your work, especially when dealing with multiple steps. Go back through each statement and make sure your explanation matches your True/False answer. Did you correctly calculate the elements of set C? Did you accurately interpret the set notation symbols? Catching small errors can make a big difference!

Key Takeaways

• Understanding Set Notation: This problem really highlights the importance of understanding set notation (∈, ∉, ⊂, etc.). Make sure you're comfortable with these symbols.
• Systematic Approach: Breaking down the problem into smaller steps (enumerating C, then analyzing each statement) makes it much easier to solve.
• Careful Calculation: Simple arithmetic errors can throw off your entire answer. Double-check your calculations!

So there you have it! We've successfully enumerated set C and analyzed a bunch of statements about these sets. I hope this explanation was clear and helpful. Keep practicing with sets, and you'll become a math whiz in no time! Good luck, guys!