ainster31 said:
tiny-tim said:hi ainster31!
let's reverse it, as you say, and then analyse it …
$$ (A\cup C)\cap B^{ C }=(A\cup C)\cap B^{ C }\\ (A\cap B^{ C })\cup (C\cap B^{ C })=(A\cup C)\cap B^{ C }\\(A-B)\cup (C-B)=(A\cup C)-B$$
from the first line to the second is the distributive rule
from the second to the third is simply applying the definition of "minus"
(but the best way would be to start from bottom left, go up the left side, and cme down the right side (or vice versa))
To prove an identity with sets, you need to show that the two sets on either side of the equal sign contain the same elements. This can be done by using the properties of sets, such as subset, superset, and equality. You can also use logical reasoning and algebraic manipulation to show that the two sets are equivalent.
Proving an identity involves showing that two sets are equivalent, while proving a theorem involves showing that a statement is true for all cases or under certain conditions. Proving an identity often requires more specific techniques, while proving a theorem may require a more general approach.
Some common techniques used to prove identities include using set properties, mathematical induction, direct proof, proof by contradiction, and proof by contrapositive. Different techniques may be more suitable depending on the specific identity being proven.
One example of proving an identity with sets is showing that the union of two sets is equal to the intersection of their complements. This can be proven using set properties, such as De Morgan's laws, and logical reasoning.
Proving identities with sets is important because it allows us to validate mathematical statements and ensure their accuracy. It also helps us to better understand the properties and relationships between different sets, and can be useful in solving more complex mathematical problems.