Question
prove that for sets SS, AA, BB, and CC, where AA, BB, and CC are subsets of SS, the following equality holds: (A−B)−C=(A−C)−(B−C)
207
likes1033 views
Answer to a math question prove that for sets SS, AA, BB, and CC, where AA, BB, and CC are subsets of SS, the following equality holds: (A−B)−C=(A−C)−(B−C)
Miles
4.9114 Answers
To prove the given equality (A - B) - C = (A - C) - (B - C), we need to show that both sides are subsets of each other.
Let x be an arbitrary element in (A - B) - C.
This means that x is in (A - B) and not in C.
To be in (A - B), x must be in A and not in B.
So, x is in A and not in B, and x is not in C.
Now, let's consider (A - C) - (B - C).
Let y be an arbitrary element in (A - C) - (B - C).
This means that y is in (A - C) and not in (B - C).
To be in (A - C), y must be in A and not in C.
To not be in (B - C), y must either not be in B or be in C.
Since y is not in C, it follows that y is not in B.
Therefore, y is in A and not in B, and y is not in C.
Thus, (A - B) - C is a subset of (A - C) - (B - C).
To show the reverse inclusion, we can follow a similar argument.
Let z be an arbitrary element in (A - C) - (B - C).
This means that z is in A and not in C, and z is not in B or is in C.
Since z is not in B or is in C, it follows that z is not in B.
Therefore, z is in (A - B) and not in C.
Hence, (A - C) - (B - C) is a subset of (A - B) - C.
Since both sides of the equality are subsets of each other, we can conclude that (A - B) - C = (A - C) - (B - C).
Answer: (A - B) - C = (A - C) - (B - C)
Let x be an arbitrary element in (A - B) - C.
This means that x is in (A - B) and not in C.
To be in (A - B), x must be in A and not in B.
So, x is in A and not in B, and x is not in C.
Now, let's consider (A - C) - (B - C).
Let y be an arbitrary element in (A - C) - (B - C).
This means that y is in (A - C) and not in (B - C).
To be in (A - C), y must be in A and not in C.
To not be in (B - C), y must either not be in B or be in C.
Since y is not in C, it follows that y is not in B.
Therefore, y is in A and not in B, and y is not in C.
Thus, (A - B) - C is a subset of (A - C) - (B - C).
To show the reverse inclusion, we can follow a similar argument.
Let z be an arbitrary element in (A - C) - (B - C).
This means that z is in A and not in C, and z is not in B or is in C.
Since z is not in B or is in C, it follows that z is not in B.
Therefore, z is in (A - B) and not in C.
Hence, (A - C) - (B - C) is a subset of (A - B) - C.
Since both sides of the equality are subsets of each other, we can conclude that (A - B) - C = (A - C) - (B - C).
Answer: (A - B) - C = (A - C) - (B - C)
Frequently asked questions (FAQs)
Math question: What is the vertex of the parabola given by the function 𝑦 = 2𝑥² + 4𝑥 - 1?
+
Math question: What is the highest point on the curve of the function f(x) = x^3 - 6x^2 + 9x within the interval [-3, 5]?
+
Math question: What is the maximum value of the graphed quadratic function y = -2x^2 + 4x - 1 within the given domain? (
+
New questions in Mathematics