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Concept · Sets & Inequalities
Set operations and relationships
Practice this concept→
A Sets & Inequalities set, untimed
Named traps
8 ways this goes wrong.
Each is a named misconception the questions are built to catch. Open one for the move that tempts, and the fix.
- Complement against wrong universeComputes a complement relative to the wrong reference/universal set.→
- Element-of vs subset-ofConfuses membership (element belongs to set) with inclusion (set is a subset).→
- Empty set as subset forgottenForgets the empty set is a subset of every set (e.g. omits the B=empty case in subset-count problems).→
- Intersection/union swappedLists union elements when intersection is required, or vice versa.→
- Nested multiple-sets not nestedFails to see multiples-of-6 are a subset of multiples-of-3 (nested parametrized integer sets).→
- Proper subset vs subsetTreats the proper-subset relation as ordinary subset-or-equal (or vice versa), mishandling the equal case.→
- Subset direction reversedReads the subset/proper-subset symbol backwards, asserting the superset is contained in the subset.→
- {empty} mistaken for empty setTreats the singleton {emptyset} or {0} as the empty set itself.→
Key formulas
12- UnionA ∪ B = { x : x ∈ A or x ∈ B }
- IntersectionA ∩ B = { x : x ∈ A and x ∈ B }
- Complement (relative to universe U)Ac = { x ∈ U : x ∉ A }
- DifferenceA − B = { x : x ∈ A and x ∉ B }
- Inclusion-exclusionn(A ∪ B) = n(A) + n(B) − n(A ∩ B)
- Subset count of an n-element set2n subsets2n − 1 proper subsets2n − 1 nonempty subsets
- Empty set ∅ is a subset of every set
- Quadratic inequality ax2+bx+c>0solution keyed to root(s) and sign of a / discriminant Δ=b2−4ac (Δ>0 two roots, Δ=0 one, Δ<0 none)
- Roots straddle k (one >k, one <k) for a>0f(k) < 0
- Rational inequality f(x)/g(x) ≤ 0sign chart on numerator/denominator zeros (g(x) ≠ 0)
- Absolute value|x − a| < b ⇔ a − b < x < a + b (b>0)
- Inequality multiplicative propertymultiplying/dividing by a negative reverses the inequality
Shaky underneath?
If the concept underneath still feels loose, patch it here before coming back to CSCA questions.
FoundationsAlgebra 2 foundations