Translate membership into logic and use double inclusion.
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Try
Two set expressions are equal when they have precisely the same members. Drawing overlapping circles suggests identities, but proving one requires tracking an arbitrary object, including cases that a convenient drawing may hide.
Predict
Predict whether “outside B and C” means outside each set or outside their intersection. Replace ambiguous prose with membership conditions. A\B means inside A and outside B; B∪C requires at least one membership; B∩C requires both.
Experiment
Use a finite universe with four objects and create membership columns for A, B and C. Compute each side of a proposed identity object by object. A mismatch supplies a counterexample; a matching finite table is evidence only for those chosen sets unless you analyze all possible membership patterns.
Understand
Worked argument: A∩(B∪C)=(A∩B)∪(A∩C). If x lies on the left, x is in A and is in B or C. In the first case it lies in A∩B; in the second it lies in A∩C, so it lies on the right. Conversely, a member of the right lies in A and in at least one of B,C, hence on the left. Both inclusions give equality.
An invalid proof stops after the first paragraph. A subset can be proper: the set containing only 1 is a subset of the set containing 1 and 2, but they are not equal. Equality requires the reverse inclusion too.
Apply
In each written task, introduce an arbitrary member and follow its conditions. Do not confuse an object belonging to A with a set being a subset of A. Explicit finite examples are welcome when disproving a false identity, provided both expressions are evaluated.
Continue
Depth: derive De Morgan’s laws relative to a specified universe. Complements depend on that universe, whereas the difference A\B explicitly names the set from which objects are removed. Explain how a change of universe affects complements but not the elements of a fixed difference.
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Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
For A={1,2,3,4}, B={2,4}, C={3,4}, enter A\(B∪C).
Integers, fractions and finite decimals are accepted. Separate values with commas; type none for no solutions. Expressions such as sqrt(2) are not parsed by this checker.
Membership in (A\B)∩(A\C) means:
Prove A\(B∪C)=(A\B)∩(A\C) by double inclusion.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Disprove A\(B∪C)=(A\B)∪(A\C) using explicit finite sets.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
