If A,B and C are three sets such that `A cap B = A cap C and A cup B = A cup C ` then
A
A=B
B
A=C
C
B=C
D
`A cap B = phi `
Text Solution
AI Generated Solution
The correct Answer is:
To solve the problem, we need to analyze the given conditions about the sets A, B, and C. The conditions state that:
1. \( A \cap B = A \cap C \)
2. \( A \cup B = A \cup C \)
We need to determine the implications of these conditions.
### Step-by-Step Solution:
**Step 1: Understand the implications of the intersections.**
From the first condition, \( A \cap B = A \cap C \), we can infer that the elements common to both A and B are the same as the elements common to A and C. This means that any element that is in both A and B must also be in C, and vice versa.
**Hint:** Think about what it means for two sets to have the same intersection with a third set.
---
**Step 2: Analyze the unions.**
From the second condition, \( A \cup B = A \cup C \), we can infer that the union of A with B is the same as the union of A with C. This means that any element that is in either A or B must also be in either A or C, and vice versa.
**Hint:** Consider how the union of sets works and what it implies about the elements in the sets.
---
**Step 3: Combine the implications.**
Since \( A \cap B = A \cap C \), we can denote this common intersection as \( X \). Thus, we can express B and C in terms of A and X:
- \( B = (A \cap B) \cup (B - A) = X \cup (B - A) \)
- \( C = (A \cap C) \cup (C - A) = X \cup (C - A) \)
Since \( A \cup B = A \cup C \), we can also express this as:
- \( A \cup (X \cup (B - A)) = A \cup (X \cup (C - A)) \)
This implies that \( B - A = C - A \).
**Hint:** Think about how the elements outside of A in B and C must relate to each other.
---
**Step 4: Conclude the relationship between B and C.**
From the previous steps, we have established that:
- The intersection of A with B and C is the same.
- The elements outside of A in B and C are also the same.
Thus, we can conclude that \( B = C \).
**Hint:** Reflect on the definitions of set equality and how the elements relate to each other.
---
### Final Conclusion:
Given the conditions \( A \cap B = A \cap C \) and \( A \cup B = A \cup C \), we conclude that \( B = C \).
To solve the problem, we need to analyze the given conditions about the sets A, B, and C. The conditions state that:
1. \( A \cap B = A \cap C \)
2. \( A \cup B = A \cup C \)
We need to determine the implications of these conditions.
### Step-by-Step Solution:
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CENGAGE ENGLISH-SET THEORY AND REAL NUMBER SYSTEM -Archives
