Establish an equivalence relation, build its classes and assemble a reviewable proof portfolio.
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An equivalence relation treats objects as the same for one specified purpose. Congruence groups integers by remainder, while equality of absolute value groups each nonzero integer with its negative. Neither relation says that the original integers are literally equal.
Predict
Predict which of reflexivity, symmetry and transitivity holds for “a is greater than b.” Reflexivity requires every object relate to itself; symmetry reverses any related pair; transitivity joins two related pairs. These are three separate proof obligations.
Experiment
Make a table for integers 0 through 7 and the relation “same remainder on division by 4.” Reorder rows to reveal the groups. Then include negative integers and identify their classes. A finite table illustrates the structure; the integer-witness proof must account for every integer.
Understand
Worked argument: on integers, define aRb when a+b is even. Reflexivity follows because a+a=2a. Symmetry follows because b+a=a+b. For transitivity, suppose a+b=2r and b+c=2s for integers r,s. Then a+c=2r+2s−2b=2(r+s−b), which is even. Thus R is an equivalence relation, with an even class and an odd class.
An invalid transitivity proof uses the same witness r for both premises. Their sums can be different even integers. Separate witnesses preserve the actual hypotheses and still yield the required conclusion.
Apply
Your final portfolio should state a fresh relation, prove its three properties and describe a class. Include one rejected proof step and the revision that repairs it. Keep a record of whether a reference solution was revealed. A saved draft or exact practice answer is not evidence of independent proof review.
Continue
Depth: equivalence classes partition the domain: every object belongs to its own class, and two classes that overlap are equal. Prove the second claim using symmetry and transitivity. Quotient sets of classes are a foundation for abstract algebra and topology; defining operations on them later requires a separate well-definedness argument.
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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.
Among integers 0 through 11, enter those equivalent to 2 when a~b means 4 divides a−b.
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.
For aRb defined by a−b>0 on integers, which statement is true?
Prove a~b iff 4 divides a−b is an equivalence relation on integers; describe its classes.
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.
Propose a different relation on integers that is an equivalence relation. Give a full proof, describe a class, and explain how finite testing and reviewer feedback affect your evidence.
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.
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