Set Theory with Three Sets and Cartesian Products

Understanding Three Sets: Regions and Operations

• Number of Regions: When dealing with three sets (A, B, and C) within a universe, there are $8$ distinct regions possible in a Venn diagram.

  • These regions can be labeled $1, 2, 3, 4, 5, 6, 7, 8$. (Note: The transcript uses $1$ for the region outside A, B, and C, and then numbers the regions within the sets). For clarity, let's consider the standard numbering where $1-7$ are within the sets and $8$ is the complement of their union.

• Identifying Regions for Individual Sets:

  • $A$ contains regions: $2, 4, 5$
  • $B$ contains regions: $2, 3, 5, 6$
  • $C$ contains regions: $4, 5, 6, 7$

• Intersection Operations (Elements in Common):

  • $A ext{ intersect } B$ ($A ext{ and } B$): regions $2, 5$ (elements common to A and B)
  • $A ext{ intersect } C$ ($A ext{ and } C$): regions $4, 5$ (elements common to A and C)
  • $B ext{ intersect } C$ ($B ext{ and } C$): regions $5, 6$ (elements common to B and C)
  • $A ext{ intersect } B ext{ intersect } C$ ($A ext{ and } B ext{ and } C$): region $5$ (elements common to all three sets)

• Union Operations (Elements in One OR the Other):

  • $A ext{ union } B$ ($A ext{ or } B$): regions $1, 2, 3, 4, 5, 6$ (elements in A or B or both)
  • $A ext{ union } C$ ($A ext{ or } C$): regions $1, 4, 5, 6, 7$ (The response in the transcript was $1, 4, 5, 6, 7$. Another suggestion was $3, 4, 5, 6, 7$. It seems the numbering for A from the transcript starts with $2,4,5$ and A and B being $1,2,3,4,5,6$ suggests a region outside A and B is included. Assuming a standard interpretation where region 1 is only A, region 2 is A and B not C, etc., then the regions given for A union B would be the numbers within sets A and B).
  • $A ext{ union } B ext{ union } C$ ($A ext{ or } B ext{ or } C$): regions $1, 2, 3, 4, 5, 6, 7$ (all regions covered by at least one of the sets)

• Combined Set Operations and Importance of Grouping Symbols:

  • $A - (B ext{ union } C)$: regions $1$ (elements only in A, neither in B nor C).
    • This can be broken down: Find $B ext{ union } C$ (regions $2, 3, 4, 5, 6, 7$), then find what is in $A$ but not in that union. If A is $2, 4, 5$, then $A - (B ext{ union } C)$ would be none if assuming the numbering from the transcript. Reinterpreting the original example's regions: if A is just region '1' (meaning only A, not B, not C), then $A - (B ext{ union } C) = 1$.
  • $A ext{ intersect } (B ext{ union } C)$: regions $2, 4, 5$
    • Steps: Identify $B ext{ union } C$ (regions $2, 3, 4, 5, 6, 7$). Then, find the intersection of $A$ (regions $2, 4, 5$) with this union. The common regions are $2, 4, 5$. This represents the part of A that overlaps with B or C.
  • $(A ext{ intersect } B) ext{ union } C$: regions $2, 4, 5, 6, 7$
    • Steps: Identify $A ext{ intersect } B$ (region $2, 5$). Identify $C$ (regions $4, 5, 6, 7$). Take the union of these two results, yielding $2, 4, 5, 6, 7$.
  • Significance: Comparing $A ext{ intersect } (B ext{ union } C)$ (regions $2, 4, 5$) with $(A ext{ intersect } B) ext{ union } C$ (regions $2, 4, 5, 6, 7$) demonstrates that grouping symbols (parentheses) matter and change the resulting set.

• Visualizing Set Operations with Colors:

  • An alternative method involves using colors in a Venn Diagram.
  • Example: For $A ext{ intersect } (B ext{ union } C)$, if $B$ is red and $C$ is blue, $B ext{ union } C$ is anything that is red or blue. Then, you find the intersection of A with this combined red/blue area.

Cardinality of Sets: Greatest and Least Number of Elements

• Problem: Given $|A| = 5$ and $|B| = 4$. Determine the greatest and least number of elements for $|A ext{ intersect } B|$ and $|A ext{ union } B|$.

• Case 1: B is a Proper Subset of A

  • This is possible since $|B| = 4$ and $|A| = 5$. Every element in B is also in A.
  • Venn Diagram: All elements of B are nested within A.
    • Region for $B$ (elements in B that are also in A): $4$ elements.
    • Region for $A$ only (elements in A but not B): $1$ element (since $5 - 4 = 1$).
  • $|A ext{ intersect } B| = 4$ (the size of set B, as B is entirely within A).
  • $|A ext{ union } B| = 5$ (the size of set A, as B adds no new elements).

• Case 2: A and B are Disjoint

  • This means A and B share nothing in common (e.g., a moon rock and a dog).
  • Venn Diagram: Two separate, non-overlapping circles.
    • Region for $A$ only: $5$ elements.
    • Region for $B$ only: $4$ elements.
    • Intersection region: $0$ elements.
  • $|A ext{ intersect } B| = 0$. This represents the least number of elements in the intersection.
  • $|A ext{ union } B| = 5 + 4 = 9$. This represents the greatest number of elements in the union.

• Case 3: A and B Share Some Elements (Intermediate Overlap)

  • If they share one element: $|A ext{ intersect } B| = 1$. Then $|A ext{ union } B| = 5 + 4 - 1 = 8$.
  • If they share two elements: $|A ext{ intersect } B| = 2$. Then $|A ext{ union } B| = 5 + 4 - 2 = 7$. (In general, $|A ext{ union } B| = |A| + |B| - |A ext{ intersect } B|$).

• Summary of Greatest and Least Values:

  • Greatest $|A ext{ intersect } B|$: $4$ (when B is a subset of A).
  • Least $|A ext{ intersect } B|$: $0$ (when A and B are disjoint).
  • Greatest $|A ext{ union } B|$: $9$ (when A and B are disjoint).
  • Least $|A ext{ union } B|$: $5$ (when B is a subset of A).

Application Examples from Textbook (Page 69)

• Example 1: Math and English Students

  • $M$ = set of students taking Mathematics.
  • $E$ = set of students taking English.
  • Venn Diagram Regions interpretation:
    • Region 'a' (or $M - E$): Students taking Mathematics but not English.
    • Region 'b' (or $M ext{ intersect } E$): Students taking Mathematics and English.
    • Region 'c' (or $E - M$): Students taking English but not Mathematics.
    • Region 'd' (or $(M ext{ union } E)^c$): Students taking neither Mathematics nor English.

• Example 2: Physics, Biology, and Math (110 Freshmen)

  • $110$ freshmen surveyed initially.
  • Given: $|P| = 25$, $|B| = 45$, $|M| = 48$.
    • Observation: The sum of these individual counts ($25 + 45 + 48 = 118$) exceeds the total number of freshmen surveyed ($110$), indicating that some students were counted more than once (i.e., they took more than one subject).
  • Given Overlaps: $|P ext{ intersect } M| = 10$ (Physics and Math), $|P ext{ intersect } B| = 8$ (Physics and Biology), $|B ext{ intersect } M| = 8$ (Biology and Math).
  • Given Triple Overlap: $|P ext{ intersect } B ext{ intersect } M| = 5$ (Physics and Biology and Math).
  • Filling the Venn Diagram (Working from the inside out):
    1. All Three (P and B and M): Place $5$ in the center region (region $P ext{ intersect } B ext{ intersect } M$).
    2. Physics and Math ONLY: $|P ext{ intersect } M| = 10$. Since $5$ are already counted in all three, the number taking Physics and Math only (not Biology) is $10 - 5 = 5$. Place $5$ in region $(P ext{ intersect } M) - B$.
    3. Biology and Math ONLY: $|B ext{ intersect } M| = 8$. Since $5$ are already counted, the number taking Biology and Math only (not Physics) is $8 - 5 = 3$. Place $3$ in region $(B ext{ intersect } M) - P$.
    4. Physics and Biology ONLY: $|P ext{ intersect } B| = 8$. Since $5$ are already counted, the number taking Physics and Biology only (not Math) is $8 - 5 = 3$. Place $3$ in region $(P ext{ intersect } B) - M$.
    5. Physics ONLY: $|P| = 25$. Those in Physics and also in other subjects are counted already in $P ext{ intersect } B$ (3), $P ext{ intersect } M$ (5), and $P ext{ intersect } B ext{ intersect } M$ (5). Total counted within P's overlaps: $3 + 5 + 5 = 13$. So, Physics only is $25 - 13 = 12$. Place $12$ in region $P - (B ext{ union } M)$.
    6. Biology ONLY: $|B| = 45$. Those in Biology and also in other subjects are counted already: $3 + 5 + 3 = 11$. So, Biology only is $45 - 11 = 34$. Place $34$ in region $B - (P ext{ union } M)$.
    7. Math ONLY: $|M| = 48$. Those in Math and also in other subjects are counted already: $5 + 5 + 3 = 13$. So, Math only is $48 - 13 = 35$. Place $35$ in region $M - (P ext{ union } B)$.
    8. Neither (Outside all three sets): Sum all the distinct regions: 12 ( ext{P only}) + 34 ( ext{B only}) + 35 ( ext{M only}) + 3 ( ext{P&B only}) + 5 ( ext{P&M only}) + 3 ( ext{B&M only}) + 5 ( ext{P&B&M}) = 97. The total surveyed was $110$. So, the number taking neither subject is $110 - 97 = 13$. Place $13$ in the region outside all three circles.

Cartesian Coordinate System and Cartesian Product

• Coordinates and Ordered Pairs:

  • Points in space are located using coordinates or ordered pairs, e.g., $(4,1)$ and $(1,4)$.
  • In 3D space, this extends to $(x,y,z)$ like $(4,1,1)$.

• Rene Descartes:

  • French philosopher and mathematician responsible for the Cartesian coordinate system.
  • Famous quote: