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.
Analyzing Set Theory Identities: Complements and Set Difference
This solution examines two statements involving set differences and complements to determine their correctness.
Statement I: Verifying $A - B = Bc - Ac$
Let's analyze the first statement: $A - B = Bc - Ac$.
-
Definition of Set Difference: The set difference $A - B$ contains all elements that are in set $A$ but not in set $B$. Mathematically, $A - B = {x | x ∈ A and x ∉ B}$.
-
Analyzing the Left Side: The left side is $A - B$, which represents elements in $A$ and not in $B$.
-
Analyzing the Right Side: The right side is $Bc - Ac$.
- Using the definition of set difference, $Bc - Ac = {x | x ∈ Bc and x ∉ Ac}$.
- Recall the definition of a complement: $x ∈ Bc$ means $x$ is not in $B$ ($x ∉ B$).
- Also, $x ∉ Ac$ means $x$ is in $A$ ($x ∈ A$).
- Substituting these back, we get: $Bc - Ac = {x | x ∉ B and x ∈ A}$.
-
Comparison: We see that ${x | x ∈ A and x ∉ B}$ is exactly the same as ${x | x ∉ B and x ∈ A}$. Therefore, the left side $A - B$ is equal to the right side $Bc - Ac$.
Conclusion for Statement I: Statement I is correct.
Statement II: Verifying $A - Bc = Ac - B$
Now, let's analyze the second statement: $A - Bc = Ac - B$.
-
Analyzing the Left Side: $A - Bc$.
- Using the definition of set difference: $A - Bc = {x | x ∈ A and x ∉ Bc}$.
- Since $x ∉ Bc$ means $x ∈ B$, we can rewrite this as: $A - Bc = {x | x ∈ A and x ∈ B}$.
- This is the definition of the intersection of $A$ and $B$, denoted as $A ∩ B$.
-
Analyzing the Right Side: $Ac - B$.
- Using the definition of set difference: $Ac - B = {x | x ∈ Ac and x ∉ B}$.
- Since $x ∈ Ac$ means $x ∉ A$, we can rewrite this as: $Ac - B = {x | x ∉ A and x ∉ B}$.
- This represents elements that are in neither $A$ nor $B$.
-
Comparison: We found that the left side equals $A ∩ B$ and the right side equals ${x | x ∉ A and x ∉ B}$. These two sets are generally not equal. For example, elements in the intersection ($A ∩ B$) are in both $A$ and $B$, while the elements described by the right side are in neither.
Counterexample: Let the universal set be $U = {1, 2, 3, 4}$, $A = {1, 2}$, and $B = {2, 3}$. Then $Ac = {3, 4}$ and $Bc = {1, 4}$.
- Left side: $A - Bc = {1, 2} - {1, 4} = {2}$.
- Right side: $Ac - B = {3, 4} - {2, 3} = {4}$.
- Since ${2} ≠ {4}$, the statement is false.
Conclusion for Statement II: Statement II is incorrect.
Final Conclusion
Based on the analysis of both statements:
- Statement I ($A - B = Bc - Ac$) is correct.
- Statement II ($A - Bc = Ac - B$) is incorrect.
Therefore, only Statement I is correct.
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?
