Identity Proof: (A-B)-C=A-(B∪C)

In summary, the concept of identity proof in mathematics states that when a number is added to or multiplied by zero, the result is always that same number. This applies to both numbers and sets, where the sum or product of any number and zero is equal to that number and the union or intersection of a set with the empty set is always equal to that original set. The identity property can be used to simplify equations and prove the equality of sets, making mathematical manipulations easier and serving as a fundamental concept in understanding numbers and sets. It is also important in more complex mathematical concepts and theories.
  • #1
gicm
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0
Show that(A-B)-C=A-(BUC)
 
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  • #2
This can be shown using identities on sets. Two identities are: $A-B=A\bar{B}$ and $\overline{B\cup C}=\bar{B}\bar{C}$. Here $\bar{A}$ denotes the complement of $A$, and I skip intersection, i.e., I write $AB$ for $A\cap B$. There are numerous other identities on sets, such as commutativity and associativity of intersection and union, laws involving the empty set and so on. Can you use the ones I provided to prove your equality?
 

1. What is "Identity Proof: (A-B)-C=A-(B∪C)"?

This is a mathematical identity that states that the difference between sets A and B, with the additional removal of set C, is equal to the difference between sets A and the union of sets B and C.

2. How is this identity useful in mathematics?

This identity is useful in simplifying and solving equations involving sets. It can also be used to prove other mathematical theorems and identities.

3. Can you provide an example of using this identity?

Sure, let's say A = {1, 2, 3}, B = {2, 3, 4}, and C = {3, 4, 5}. Using the identity, we can see that (A-B)-C = (1) - {3} = {1}, and A-(B∪C) = {1, 2} - {2, 3, 4, 5} = {1}. Therefore, (A-B)-C = A-(B∪C) and the identity holds.

4. Are there any exceptions to this identity?

No, this identity holds true for all possible combinations of sets A, B, and C.

5. Can this identity be used in other areas of science?

Yes, this identity can also be applied in fields such as computer science, statistics, and economics, where sets and set operations are commonly used.

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