- 1. An Operation Is a Rule for Two Inputs
Read a binary operation defined by a rule, evaluate it by substituting with brackets, see that the order of the inputs matters, and evaluate nested expressions and expressions with two operations.
- 2. Solving Equations Built from an Operation
Turn a statement such as 5 * x = 37 into an equation, solve for the missing input in either slot, handle an operation on each side or two operations in one equation, and solve quadratics that come from an operation, keeping and checking both answers.
- 3. Does It Always Work? Closure and Commutativity
Decide whether an operation is commutative or closed: one counter-example disproves a property, examples that agree prove nothing, swapping the letters proves commutativity, the set is half of closure, and operation tables show both at a glance.
- 4. Identity and Inverse Elements
Find the identity element of an operation (one number that works on both sides), find the inverse of an element from a * x = e, spot the element with no inverse, and read the identity and inverses from an operation table.
- 5. Associativity: Regroup, Don't Reorder
Test associativity by working out both groupings of a triple, prove or disprove it by expanding (a*b)*c and a*(b*c), see that associative and commutative are different questions, and use the symmetric-expansion shortcut for commutative rules.
- 6. Distributivity and the Laws of Sets
Test whether one operation is distributive over another (left and right, by a triple or by expanding), see that multiplication distributes over addition but not conversely, learn the commutative, associative and distributive laws of sets, and name a property from its equation.
- 7. Putting It All Together: Deciding Every Property
Run the five-question check (closed, commutative, associative, identity, inverse) on any operation with the right evidence, see how the set and one failed test decide the verdict, read the properties from a table, and spot the planted mistakes.
