The Cartesian product A × A has 16 elements among which are (0, 2) and (1, 3). Which of the following statements is/are correct? 1. It is possible to determine set A. 2. A × A contains the element (3, 2). Select the correct answer using the code given below:
Both 1 and 2
Understanding the Cartesian Product
The question asks about a Cartesian product A × A. The Cartesian product of two sets A and B, denoted by A × B, is the set of all ordered pairs (a, b) where 'a' is an element of A, and 'b' is an element of B. In this case, the product is A × A, meaning it's the set of all ordered pairs (a, b) where both 'a' and 'b' are elements of the set A.
The number of elements in the Cartesian product A × A is given as 16. The cardinality (number of elements) of a Cartesian product is the product of the cardinalities of the individual sets. So, |A × A| = |A| × |A| = |A|2.
Given that |A × A| = 16, we have |A|2 = 16.
To find the cardinality of set A, we take the square root of 16. Since the cardinality must be a non-negative integer, |A| = \(\sqrt{16} = 4\). So, set A has exactly 4 elements.
Analyzing Statement 1: Possibility to Determine Set A
Statement 1 says it is possible to determine set A. We know |A| = 4. We are also given that the elements (0, 2) and (1, 3) are present in A × A.
According to the definition of the Cartesian product A × A, if an ordered pair (x, y) is in A × A, then 'x' must be an element of A, and 'y' must be an element of A.
- Since (0, 2) \(\in\) A × A, it means that 0 \(\in\) A and 2 \(\in\) A.
- Since (1, 3) \(\in\) A × A, it means that 1 \(\in\) A and 3 \(\in\) A.
From these two given elements, we know that the numbers 0, 1, 2, and 3 must all be elements of set A. So, the set {0, 1, 2, 3} is a subset of A ({0, 1, 2, 3} \(\subseteq\) A).
We previously determined that the cardinality of A is 4 (|A| = 4). We have found 4 distinct elements (0, 1, 2, and 3) that must be in A. Since A contains exactly 4 elements and we have identified 4 elements that it must contain, these must be the only elements in A.
Therefore, set A must be {0, 1, 2, 3}. We have successfully determined set A.
Statement 1 is correct.
Analyzing Statement 2: A × A Contains Element (3, 2)
Statement 2 asks if A × A contains the element (3, 2). Based on our analysis of Statement 1, we determined that the set A is {0, 1, 2, 3}.
The Cartesian product A × A is formed by taking all possible ordered pairs where the first element comes from A and the second element comes from A.
A × A = {(a, b) | a \(\in\) A and b \(\in\) A}
Using A = {0, 1, 2, 3}, the elements of A × A are:
| First Element (from A) | Second Element (from A) | Ordered Pair |
|---|---|---|
| 0 | 0, 1, 2, 3 | (0, 0), (0, 1), (0, 2), (0, 3) |
| 1 | 0, 1, 2, 3 | (1, 0), (1, 1), (1, 2), (1, 3) |
| 2 | 0, 1, 2, 3 | (2, 0), (2, 1), (2, 2), (2, 3) |
| 3 | 0, 1, 2, 3 | (3, 0), (3, 1), (3, 2), (3, 3) |
To check if (3, 2) is in A × A, we need to see if the first element, 3, is in A and if the second element, 2, is in A.
From A = {0, 1, 2, 3}, we see that:
- 3 \(\in\) A (True)
- 2 \(\in\) A (True)
Since both elements 3 and 2 are in set A, the ordered pair (3, 2) is indeed an element of A × A.
Statement 2 is correct.
Conclusion on Statements
Both Statement 1 and Statement 2 are correct.
- Statement 1 is correct because knowing the cardinality of A × A and two specific elements allowed us to deduce the unique set A.
- Statement 2 is correct because once we determined A = {0, 1, 2, 3}, we could confirm that (3, 2) is an element of A × A as both 3 and 2 belong to A.
Revision Table: Key Concepts
| Concept | Explanation | Application in Problem |
|---|---|---|
| Cartesian Product (A × B) | Set of all ordered pairs (a, b) where a \(\in\) A, b \(\in\) B. | Used to define A × A and its elements. |
| Cardinality (|S|) | Number of elements in a set S. | \(|\)A × A\(|\) = \(|\)A\(|^2\); Used to find \(|\)A\(|\). |
| Elements of A × A | If (x, y) \(\in\) A × A, then x \(\in\) A and y \(\in\) A. | Used to deduce elements of A from (0, 2) and (1, 3). |
| Determining Set A | Knowing \(|\)A\(|\) and identifying all its elements. | \(|\)A\(|\)=4 and {0, 1, 2, 3} \(\subseteq\) A \(\implies\) A = {0, 1, 2, 3}. |
Additional Information: Properties of Cartesian Products
- The order of elements in an ordered pair matters, i.e., (a, b) is generally not equal to (b, a) unless a = b.
- If A or B is an empty set (\(\emptyset\)), then A × B is also an empty set (\(\emptyset\)).
- If A and B are finite sets, then \(|\)A × B\(|\) = \(|\)A\(|\) \(\times\) \(|\)B\(|\).
- The Cartesian product is not commutative, i.e., A × B \(\neq\) B × A unless A = B or A or B is empty. In our case, A × A = A × A is trivially true because it's the same product.
- The Cartesian product can be extended to more than two sets, e.g., A × B × C would be a set of ordered triples (a, b, c).
Similar Questions
If A = {{1, 2, 3}}, then how many elements are there in the power set of A?
Consider the following statements in respect of two non-empty sets A and B :
1. x ∉ (A ∪ B) ⇒ x ∉ A or x ∉ B
2. x ∉ (A ∩ B) ⇒ x ∉ A and x ∉ B
Which of the above statements is/are correct?
Consider the proper subsets of {1, 2, 3, 4}. How many of these proper subsets are a superset of the set {3}?
- Set \(X\) contains \(3n\) elements and set \(Y\) contains \(2n\) elements, and they have \(n\) elements in common. How many elements does \((X – Y ) \times (Y – X)\) have?
If \(A\) and \(B\) are non-empty subsets of a set, and \(A^c\) and \(B^c\) represent their complements, then which of the following is/are correct?
I. \(A-B = B^c - A^c\)
II. \(A - B^c = A^c - B\)
Select the answer using the code given below.
Consider the following statements in respect of two non-empty sets A and B :
1. A ∪ B = A ∩ B if A = B
2. A Δ B = ϕ if A = B
Which of the above statements is/are correct ?
Important Questions from Sets
Consider two subsets of ℝ 2given as, S1 = {[1, -2], [3, 5]} and S2 = {[1, 1], [0, 0]}. Then,
The standard ordered basis of ℝ 2is {e 1, e 2}. Let T : ℝ 2 → ℝ 2 be the linear transformation such that T reflects the points through the line x 1= -x 2. The standard matrix of T is:
In a class, 20 students opted for physics, 17 for Maths, 12 for both physics and maths and 10 students for other subjects. The class contains how many students?
A college awarded 38 medals in Football, 15 in Basketball and 20 in Cricket. If these medals went to a total of 58 men and only 3 men got medals in all the 3 sports, how many received medals in exactly two of the 3 sports?
The set N of natural numbers is:
