If A = {x : 0 ≤ x ≤ 2} and B = {y; y is a prime number}, then what is A ∩ B equal to?
{2}
Finding the Intersection of Sets A and B
The problem asks us to find the intersection of two sets, set A and set B. The intersection of two sets, denoted by $\text{A } \cap \text{ B}$, contains all the elements that are common to both sets A and B.
Understanding Set A
Set A is defined as $\{x : 0 \le x \le 2\}$. This notation describes a set containing all real numbers $x$ such that $x$ is greater than or equal to 0 and less than or equal to 2. In interval notation, this set is represented as $[0, 2]$.
So, Set A = $[0, 2]$.
Understanding Set B
Set B is defined as $\{y; y \text{ is a prime number}\}$. A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself.
The first few prime numbers are 2, 3, 5, 7, 11, 13, and so on.
So, Set B = $\{2, 3, 5, 7, 11, \dots\}$.
Calculating the Intersection A ∩ B
To find $\text{A } \cap \text{ B}$, we need to identify the elements that are present in both Set A and Set B. We are looking for numbers that are simultaneously in the interval $[0, 2]$ AND are prime numbers.
- Set A contains all real numbers from 0 to 2, including 0 and 2.
- Set B contains prime numbers, which are positive integers greater than 1 (2, 3, 5, ...).
Let's examine the numbers in the interval $[0, 2]$ to see which ones are also prime numbers:
- Consider the integers in the interval $[0, 2]$: These are 0, 1, and 2.
- Check if these integers are prime:
- 0: Not a positive integer greater than 1, so not prime.
- 1: Not greater than 1, so not prime.
- 2: A positive integer greater than 1. Its only positive divisors are 1 and 2. Therefore, 2 is a prime number.
Are there any other numbers in the interval $[0, 2]$ (real numbers) that are prime? Prime numbers are defined only for positive integers. So, we only need to consider the integers within the interval $[0, 2]$ that are also positive and greater than 1. The only such integer is 2.
The only number that is in both Set A (the interval $[0, 2]$) and Set B (the set of prime numbers $\{2, 3, 5, \dots\}$) is 2.
Therefore, the intersection $\text{A } \cap \text{ B}$ is the set containing only the element 2.
$\text{A } \cap \text{ B} = \{2\}$.
Analyzing the Options
Let's compare our result with the given options:
- $\Phi$: The empty set. This is incorrect because we found that 2 is in the intersection.
- $\{1\}$: This is incorrect because 1 is not a prime number.
- $\{2\}$: This matches our calculated intersection.
- $\{1, 2\}$: This is incorrect because 1 is not a prime number, although 2 is.
The correct option is $\{2\}$.
| Set | Description | Elements |
|---|---|---|
| A | Real numbers $x$ such that $0 \le x \le 2$ | $[0, 2]$ (interval) |
| B | Prime numbers | $\{2, 3, 5, 7, \dots\}$ |
| A ∩ B | Elements common to A and B | $\{2\}$ |
Revision Table: Key Concepts
| Term | Definition | Example |
|---|---|---|
| Set | A collection of distinct elements. | $\{1, 2, 3\}$ |
| Intersection ($\cap$) | The set of elements common to two or more sets. | If P={1,2}, Q={2,3}, P ∩ Q = {2} |
| Interval Notation | A way to represent a range of real numbers. | $[a, b]$ means $a \le x \le b$ |
| Prime Number | A positive integer > 1 with exactly two distinct positive divisors: 1 and itself. | 2, 3, 5, 7, 11 |
| Empty Set ($\Phi$ or {}) | A set containing no elements. | The set of prime numbers < 2. |
Additional Information: Set Operations and Prime Numbers
Understanding set operations like intersection is fundamental in set theory. Set operations allow us to combine sets or find relationships between them. Besides intersection, common set operations include union ($\cup$), difference (A - B), and complement (Ac).
- Union ($\cup$): The union of two sets A and B ($\text{A } \cup \text{ B}$) is the set containing all elements that are in A or in B or in both. For example, if A = $\{1, 2\}$ and B = $\{2, 3\}$, then $\text{A } \cup \text{ B} = \{1, 2, 3\}$.
- Set Difference (A - B): The set difference A - B (or A \ B) is the set of elements that are in A but not in B. If A = $\{1, 2, 3\}$ and B = $\{2, 4\}$, then A - B = $\{1, 3\}$.
Prime numbers are building blocks in number theory. The number 2 is unique among prime numbers because it is the only even prime number. All other prime numbers are odd.
This problem combined concepts from set theory (intervals, intersection) and number theory (prime numbers), which is common in mathematics questions.
Important Questions from Union and Intersection of Sets
Consider the following statements
1. A = (A ∪ B) ∪ (A - B),
2. A ∪ (B - A) = (A ∪ B)
3. B = (A ∪ B) - (A - B)
Which of the statements given above are correct?
If a set A contains 3 elements and another set B contains 6 elements, then what is the minimum number of elements that (A∪B) can have?
Let A ∪ B = {x|(x - a)(x - b) > 0, where a < b}. What are A and B equal to?
If A = {x ϵ Z : x 3– 1 = 0} and B = {x ϵ Z: x 2+ x + 1 = 0}, where Z is set of complex numbers, then what is A ∩ B equal to?
In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper I, 26 read newspaper T, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find the number of students who read exactly one newspaper.
