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Prove that
A intersection (B minus C) is equal to (A intersection B) minus (A intersection C)?
A intersection (B minus C) is equal to (A intersection B) minus (A intersection C)?
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Proof of A ∩ (B - C) = (A ∩ B) - (A ∩ C)
Step 1: Definition of A ∩ (B - C)
- A ∩ (B - C) represents the elements that are common in set A and the elements of set B that are not in set C.
Step 2: Definition of (A ∩ B) - (A ∩ C)
- (A ∩ B) - (A ∩ C) represents the elements that are common in both set A and set B, but not in set A and set C simultaneously.
Step 3: Showing Equality
- Let x be an arbitrary element in A ∩ (B - C).
- This means x ∈ A and x ∈ B but x ∉ C.
- So, x ∈ A and x ∈ B, which implies x ∈ A ∩ B.
- Also, x ∉ C, which means x ∉ A ∩ C.
- Therefore, x ∈ (A ∩ B) - (A ∩ C), proving A ∩ (B - C) ⊆ (A ∩ B) - (A ∩ C).
Step 4: Reverse Inclusion
- Let y be an arbitrary element in (A ∩ B) - (A ∩ C).
- This means y ∈ A, y ∈ B, but y ∉ C.
- So, y ∈ A and y ∈ B, which implies y ∈ A ∩ (B - C).
- Also, y ∉ C, which means y ∉ A ∩ C.
- Therefore, y ∈ A ∩ (B - C), proving (A ∩ B) - (A ∩ C) ⊆ A ∩ (B - C).
Step 5: Conclusion
- From steps 3 and 4, we have shown that A ∩ (B - C) ⊆ (A ∩ B) - (A ∩ C) and (A ∩ B) - (A ∩ C) ⊆ A ∩ (B - C).
- Hence, A ∩ (B - C) = (A ∩ B) - (A ∩ C) is proved.
Step 1: Definition of A ∩ (B - C)
- A ∩ (B - C) represents the elements that are common in set A and the elements of set B that are not in set C.
Step 2: Definition of (A ∩ B) - (A ∩ C)
- (A ∩ B) - (A ∩ C) represents the elements that are common in both set A and set B, but not in set A and set C simultaneously.
Step 3: Showing Equality
- Let x be an arbitrary element in A ∩ (B - C).
- This means x ∈ A and x ∈ B but x ∉ C.
- So, x ∈ A and x ∈ B, which implies x ∈ A ∩ B.
- Also, x ∉ C, which means x ∉ A ∩ C.
- Therefore, x ∈ (A ∩ B) - (A ∩ C), proving A ∩ (B - C) ⊆ (A ∩ B) - (A ∩ C).
Step 4: Reverse Inclusion
- Let y be an arbitrary element in (A ∩ B) - (A ∩ C).
- This means y ∈ A, y ∈ B, but y ∉ C.
- So, y ∈ A and y ∈ B, which implies y ∈ A ∩ (B - C).
- Also, y ∉ C, which means y ∉ A ∩ C.
- Therefore, y ∈ A ∩ (B - C), proving (A ∩ B) - (A ∩ C) ⊆ A ∩ (B - C).
Step 5: Conclusion
- From steps 3 and 4, we have shown that A ∩ (B - C) ⊆ (A ∩ B) - (A ∩ C) and (A ∩ B) - (A ∩ C) ⊆ A ∩ (B - C).
- Hence, A ∩ (B - C) = (A ∩ B) - (A ∩ C) is proved.
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Prove that A intersection (B minus C) is equal to (A intersection B) minus (A intersection C)?
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Prove that A intersection (B minus C) is equal to (A intersection B) minus (A intersection C)? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Prove that A intersection (B minus C) is equal to (A intersection B) minus (A intersection C)? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that A intersection (B minus C) is equal to (A intersection B) minus (A intersection C)?.
Prove that A intersection (B minus C) is equal to (A intersection B) minus (A intersection C)? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Prove that A intersection (B minus C) is equal to (A intersection B) minus (A intersection C)? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that A intersection (B minus C) is equal to (A intersection B) minus (A intersection C)?.
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