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 ?
Both 1 and 2
Understanding Set Operations and Properties
This question asks us to evaluate the correctness of two statements regarding operations on two non-empty sets, A and B, specifically when the sets are equal (A = B).
Let's analyze each statement separately, keeping in mind that A and B are non-empty sets and the condition is A = B.
Analyzing Statement 1: A ∪ B = A ∩ B if A = B
Statement 1 claims that the union of sets A and B is equal to the intersection of sets A and B if A is equal to B.
- The union of two sets A and B, denoted by $A \cup B$, is the set of all elements that are in A, or in B, or in both.
- The intersection of two sets A and B, denoted by $A \cap B$, is the set of all elements that are common to both A and B.
Given the condition $A = B$, let's see what $A \cup B$ and $A \cap B$ become:
- If $A = B$, then $A \cup B$ is the union of A with itself, which is simply A. So, $A \cup B = A$.
- If $A = B$, then $A \cap B$ is the intersection of A with itself, which is also simply A. So, $A \cap B = A$.
Since both $A \cup B$ and $A \cap B$ are equal to A when $A = B$, it follows that $A \cup B = A \cap B$.
Therefore, Statement 1 is correct.
Analyzing Statement 2: A Δ B = ϕ if A = B
Statement 2 claims that the symmetric difference of sets A and B is equal to the empty set ($\varphi$) if A is equal to B.
- The symmetric difference of two sets A and B, denoted by $A \Delta B$, is the set of elements which are in either A or B, but not in their intersection. It can be defined as $(A \cup B) - (A \cap B)$ or $(A - B) \cup (B - A)$.
- The empty set, denoted by $\varphi$, is the set containing no elements.
Given the condition $A = B$, let's use the definition $A \Delta B = (A - B) \cup (B - A)$:
- If $A = B$, then $A - B$ is the set of elements in A but not in B. Since A and B have exactly the same elements, there are no elements in A that are not in B. So, $A - B = \varphi$.
- Similarly, if $A = B$, then $B - A$ is the set of elements in B but not in A. Since A and B have the same elements, there are no elements in B that are not in A. So, $B - A = \varphi$.
Now, let's find the union of these two results:
- $A \Delta B = (A - B) \cup (B - A) = \varphi \cup \varphi = \varphi$.
Alternatively, using the definition $A \Delta B = (A \cup B) - (A \cap B)$: We already found that if $A=B$, then $A \cup B = A$ and $A \cap B = A$. So, $A \Delta B = A - A = \varphi$.
In both cases, the symmetric difference $A \Delta B$ is the empty set when $A = B$.
Therefore, Statement 2 is correct.
Conclusion on Set Statements
Both Statement 1 ($A \cup B = A \cap B$ if $A = B$) and Statement 2 ($A \Delta B = \varphi$ if $A = B$) have been shown to be correct based on the definitions of set union, intersection, and symmetric difference when the sets A and B are equal.
| Operation | Definition | Result when A = B | Equality Check |
|---|---|---|---|
| Union ($A \cup B$) | Elements in A or B or both | $A \cup A = A$ | |
| Intersection ($A \cap B$) | Elements common to A and B | $A \cap A = A$ | $A \cup B = A \cap B$ (Both are A) |
| Symmetric Difference ($A \Delta B$) | $(A-B) \cup (B-A)$ or $(A \cup B) - (A \cap B)$ | $(A-A) \cup (A-A) = \varphi \cup \varphi = \varphi$ OR $A - A = \varphi$ | $A \Delta B = \varphi$ |
Based on our analysis, both statements are correct.
Revision Table: Key Set Concepts
| Concept | Notation | Description |
|---|---|---|
| Union | $A \cup B$ | Elements in A or B (or both) |
| Intersection | $A \cap B$ | Elements common to A and B |
| Difference | $A - B$ | Elements in A but not in B |
| Symmetric Difference | $A \Delta B$ | Elements in A or B but not both ($ (A-B) \cup (B-A) $) |
| Empty Set | $\varphi$ or ${} $ | A set with no elements |
Additional Information: Properties of Sets
When working with sets, understanding basic properties and definitions is crucial. Here are a few more points:
- Set Equality: Two sets A and B are equal ($A = B$) if and only if they have exactly the same elements. This means every element in A is in B, and every element in B is in A.
- Non-empty Sets: The problem specifies that A and B are non-empty sets. This means they each contain at least one element. However, the results we derived ($A \cup B = A \cap B = A$ and $A \Delta B = \varphi$) when $A=B$ would also hold true even if A and B were the empty set ($\varphi$), as $\varphi = \varphi$. $\varphi \cup \varphi = \varphi$ and $\varphi \cap \varphi = \varphi$, so $\varphi \cup \varphi = \varphi \cap \varphi$. Also, $\varphi \Delta \varphi = (\varphi - \varphi) \cup (\varphi - \varphi) = \varphi \cup \varphi = \varphi$.
- Set Operations are Well-Defined: Union, intersection, and symmetric difference are fundamental operations in set theory and have clear definitions regardless of the contents of the sets.
Important Questions from Sets
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:
Consider the proper subsets of {1, 2, 3, 4}. How many of these proper subsets are a superset of the set {3}?
In a class of $200$ students numbered $1$ to $200$, those whose number is divisible by $2$ opted for Literature, those whose number is divisible by $3$ opted for History, and those whose number is divisible by $7$ opted for Philosophy. Then the number of students who did not opt for any of the three courses is:
Let $f(x) = |x - 2| + |x - 8|$; $x \in R$. Then the set of all values of $x$, at which the function, $g(x) = f(f(x))$ is not differentiable, is:
If A = {x : x is a multiple of 7},
B = {x : x is a multiple of 5} and
C = {x : x is a multiple of 35}
Then which of the following is null set?
