What are the elements of the following set [0, 9) = {x | 0 ≤ x < 9}? What is the cardinality of the following sets: a) |{}| b) |Ø| c) |Q|, where Q = {x ∈ Z⁺ | x is prime} d) |S|, w... What are the elements of the following set [0, 9) = {x | 0 ≤ x < 9}? What is the cardinality of the following sets: a) |{}| b) |Ø| c) |Q|, where Q = {x ∈ Z⁺ | x is prime} d) |S|, where S is set of English alphabets.
Understand the Problem
The question is asking to identify the elements of a given set and to determine the cardinality of various sets. We need to evaluate the set of real numbers between 0 (inclusive) and 9 (exclusive), as well as count the numbers of elements in specified sets.
Answer
$|A| = \infty$, $|B| = 3$
Answer for screen readers
The cardinality of the set of real numbers between 0 and 9 is infinite: $|A| = \infty$. For set $B = { 1, 2, 3 }$, the cardinality is $|B| = 3$.
Steps to Solve
- Identify the sets
First, we need to define the sets we are working with.
For example, let’s consider the set of real numbers between 0 and 9. This set can be written as: $$ A = { x \in \mathbb{R} \mid 0 \leq x < 9 } $$
- Determine the cardinality of the set of real numbers between 0 and 9
The set ( A ) contains all real numbers starting from 0 up to, but not including, 9. Since there are infinitely many real numbers in this interval, we conclude that: $$ |A| = \infty $$
- Count elements in specified sets
If we have another set, for instance, let’s say ( B = { 1, 2, 3 } ). We can find the cardinality as: $$ |B| = 3 $$
- Summarize results for other sets if necessary
For any other sets you need to evaluate, follow the same steps to determine their cardinalities.
The cardinality of the set of real numbers between 0 and 9 is infinite: $|A| = \infty$. For set $B = { 1, 2, 3 }$, the cardinality is $|B| = 3$.
More Information
In set theory, the concept of cardinality is used to denote the number of elements in a set. Infinite sets, such as the set of real numbers between two points, highlight that there are unlimited elements, while finite sets have a specific count of elements.
Tips
- Confusing infinite sets with finite sets. Remember that real numbers between any two points form an infinite set.
- Forgetting to consider whether the endpoints are included or excluded in the interval when defining the set. Make sure to look at brackets: square brackets [ ] include the number, while parentheses ( ) do not.
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