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Sets, Functions, and Relations Lab

Retrieval Prompts​

  1. State the definitions of subset, set equality, injective, surjective, equivalence relation, and partial order.
  2. Describe the standard proof strategy for set equality.
  3. State the difference between image and preimage.
  4. State the difference between symmetric and antisymmetric.

Compare and Distinguish​

Write a short paragraph for each pair:

  • x in A versus A subseteq B
  • codomain versus range
  • injective versus surjective
  • equivalence relation versus partial order
  • minimal element versus least element

Common Mistake Check​

Diagnose the flaw in each claim:

  1. "Because {1} in P({1,2}), therefore {1} subseteq P({1,2})."
  2. "The function f(x) = x^2 on the reals is bijective because every input has one output."
  3. "A relation is antisymmetric exactly when it is not symmetric."
  4. "To prove two sets are equal, it is enough to show the left side is contained in the right side."

Mini Application​

Complete all four tasks:

  1. Prove one nontrivial set identity by double inclusion.
  2. Classify two functions of your choice as injective, surjective, bijective, or neither, and justify each from the definitions.
  3. Build one relation on a finite set that is an equivalence relation and describe its equivalence classes.
  4. Build one relation on a finite set that is a partial order but not a total order, then identify an incomparable pair.

Evidence Check​

This page is complete only if you produced at least one proof, at least two classifications with justification, and at least one explicit counterexample to a false claim.

Integrated Set Fluency Drill​

Let U be the set of integers from 1 through 20, A the multiples of 2, B the multiples of 3, and C the primes.

Complete the worksheet:

  1. Write A, B, C, A union B, A intersect C, B - A, and U - (A union C) in roster notation.
  2. Compute the cardinality of each set and explain one result with inclusion-exclusion.
  3. List all subsets of {x, y, z} and then write its power set.
  4. Decide whether A subseteq U, C subseteq A, and {2, 3} subseteq C; justify each from the definition.
  5. Give one example where confusing element membership with subset inclusion leads to a wrong conclusion.

Evidence check: your submission must contain symbols and sentences. A correct roster without a definition-based explanation is not enough.