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A×(B intersection C)=(A×B) intersection(A×C)
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A×(B intersection C)=(A×B) intersection(A×C) Related: Sets and their ...
With B intersection C gives. you the same answer as a union B intersection. with a union C at a intersection with B Union C gives. you the same answer as a intersection B Union with a intersection C
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A×(B intersection C)=(A×B) intersection(A×C) Related: Sets and their ...
Proof of A×(B ∩ C) = (A×B) ∩ (A×C)
Definition of Cartesian Product:
- The Cartesian product A×B of two sets A and B is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B.
Proof:
- Let (x, y) be an element of A×(B ∩ C), where x is an element of A and y is an element of B ∩ C.
- This implies that y is an element of both B and C.
- Therefore, (x, y) is an element of A×B and A×C.
- This shows that (x, y) is an element of (A×B) ∩ (A×C).
- Now, let (x, y) be an element of (A×B) ∩ (A×C).
- This means that (x, y) is both in A×B and A×C.
- By the definition of Cartesian product, this implies that x is an element of A and y is an element of B and C.
- Hence, y must be in both B and C, i.e., y is in B ∩ C.
- Therefore, (x, y) is an element of A×(B ∩ C).
Conclusion:
- From the above proof, we can conclude that A×(B ∩ C) = (A×B) ∩ (A×C).
Definition of Cartesian Product:
- The Cartesian product A×B of two sets A and B is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B.
Proof:
- Let (x, y) be an element of A×(B ∩ C), where x is an element of A and y is an element of B ∩ C.
- This implies that y is an element of both B and C.
- Therefore, (x, y) is an element of A×B and A×C.
- This shows that (x, y) is an element of (A×B) ∩ (A×C).
- Now, let (x, y) be an element of (A×B) ∩ (A×C).
- This means that (x, y) is both in A×B and A×C.
- By the definition of Cartesian product, this implies that x is an element of A and y is an element of B and C.
- Hence, y must be in both B and C, i.e., y is in B ∩ C.
- Therefore, (x, y) is an element of A×(B ∩ C).
Conclusion:
- From the above proof, we can conclude that A×(B ∩ C) = (A×B) ∩ (A×C).
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A×(B intersection C)=(A×B) intersection(A×C) Related: Sets and their Representation - Sets, Class 11, Mathematics
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A×(B intersection C)=(A×B) intersection(A×C) Related: Sets and their Representation - Sets, Class 11, Mathematics for Commerce 2024 is part of Commerce preparation. The Question and answers have been prepared according to the Commerce exam syllabus. Information about A×(B intersection C)=(A×B) intersection(A×C) Related: Sets and their Representation - Sets, Class 11, Mathematics covers all topics & solutions for Commerce 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A×(B intersection C)=(A×B) intersection(A×C) Related: Sets and their Representation - Sets, Class 11, Mathematics.
A×(B intersection C)=(A×B) intersection(A×C) Related: Sets and their Representation - Sets, Class 11, Mathematics for Commerce 2024 is part of Commerce preparation. The Question and answers have been prepared according to the Commerce exam syllabus. Information about A×(B intersection C)=(A×B) intersection(A×C) Related: Sets and their Representation - Sets, Class 11, Mathematics covers all topics & solutions for Commerce 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A×(B intersection C)=(A×B) intersection(A×C) Related: Sets and their Representation - Sets, Class 11, Mathematics.
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