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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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Proof of A ∩ (B - C) = (A ∩ B) - (A ∩ C)
Explanation:
- Let's begin by expanding the left-hand side of the equation A ∩ (B - C):
- A ∩ (B - C) = {x | x ∈ A and x ∈ (B - C)}
- This can be further simplified as A ∩ (B - C) = {x | x ∈ A and x ∈ B and x ∉ C}
- Now, let's expand the right-hand side of the equation (A ∩ B) - (A ∩ C):
- (A ∩ B) - (A ∩ C) = {y | y ∈ (A ∩ B) and y ∉ (A ∩ C)}
- Breaking it down further, we get (A ∩ B) - (A ∩ C) = {y | y ∈ A and y ∈ B and y ∉ C}
- Comparing the two expressions, we see that both sides represent the set of elements that belong to A and B but not to C. Therefore, A ∩ (B - C) = (A ∩ B) - (A ∩ C).
- Thus, we have successfully proven that A ∩ (B - C) is equal to (A ∩ B) - (A ∩ C) using set theory principles.
This proof showcases the application of set operations and how they can be manipulated to derive equivalent expressions.
Explanation:
- Let's begin by expanding the left-hand side of the equation A ∩ (B - C):
- A ∩ (B - C) = {x | x ∈ A and x ∈ (B - C)}
- This can be further simplified as A ∩ (B - C) = {x | x ∈ A and x ∈ B and x ∉ C}
- Now, let's expand the right-hand side of the equation (A ∩ B) - (A ∩ C):
- (A ∩ B) - (A ∩ C) = {y | y ∈ (A ∩ B) and y ∉ (A ∩ C)}
- Breaking it down further, we get (A ∩ B) - (A ∩ C) = {y | y ∈ A and y ∈ B and y ∉ C}
- Comparing the two expressions, we see that both sides represent the set of elements that belong to A and B but not to C. Therefore, A ∩ (B - C) = (A ∩ B) - (A ∩ C).
- Thus, we have successfully proven that A ∩ (B - C) is equal to (A ∩ B) - (A ∩ C) using set theory principles.
This proof showcases the application of set operations and how they can be manipulated to derive equivalent expressions.
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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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