Introduction to Set Theory

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Misc

Mid-semester course evaluation: optional paper survey re what in this course works for you, what doesn’t, what you would like to change, etc.

• Complete by Wednesday if you want to do it.

Exam 2 solutions are available in PDF and LaTeX.

Questions?

Set Operations

Section 5.1

Example

Sets of people

Sophomores = { IC, RM, ML, AL, SN, SP, JM, JB, SO }

Juniors = { TO, VL, AM, EF, JG }

Athletes = { JG, AL, ML }

Math (math majors) = { HF, CA, RM, ML, AL, FI, SP, TO, MW, JM, AB, WF }

Universal set (students in this section of Math 239) = { HF, CA, IC, TO, RM, ML, AL, FI, SN, AH, SP, VL, AM, EF, JM, WF, MW, RS, SO, AB, JB, JG, JT }

Basic operations on these sets

Relevant ideas or questions from the reading

• Intersection = elements that are in both of 2 sets. Conjunction (and).
• Union = elements that are in either or both of 2 sets. Disjunction (or).
• Difference = all elements from one set but not other. Written A - B, where A and B are sets. Also known as relative complement.
• Complement = all elements from the universal set that aren’t in the complemented set. Written A^C, where A is the set being complemented.

Intersections

• juniors ∩ athletes = { JG }
• sophomores ∩ juniors = {}. Such sets with an empty intersection, i.e., no elements in common, are called “disjoint sets”

Unions

• juniors ∪ athletes = { TO, VL, AM, EF, JG, AL, ML }

Differences

• math - sophomores = { HF, CA, FI, TO, MW, AB, WF }
• sophomores - math = { IC, SN, JB, SO }
• Comparing these two differences suggests some conjectures that we can revisit as we learn more about proving things about sets
  • Conjecture: The intersection of the two differences is empty
  • Conjecture: The union of the two differences excludes math ∩ sophomores
  • Conjecture: For any sets A and B, (A - B) ∪ (B - A) = A ∪ B - (A∩B)

Complements

• math^C = U - math

Power set

The power set of A is the set of all subsets of A

Power set of athletes = { {}, {JG}, {AL}, {ML}, {JG,AL}, {AL,ML}, {JG,ML}, {JG,AL,ML} }

Problem Set

See handout

Next

Proving set relationships

Read textbook section 5.2

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