Arithmetics on cardinality (two sets)

Cardinality of Sets

Find all 16 cardinalities from the Venn diagram below

Score0

Correct0

Wrong0

Remaining16

🎉 Excellent Work!

You found all 16 cardinalities correctly. Click New Game to try again!

Venn Diagram

Only A (p)

A∩B (r)

Only B (q)

Outside (s)

Find the cardinality — press Enter or Check

Cardinality of Sets

What is Cardinality?

The cardinality of a set A is the number of elements in A, denoted \(n(A)\) or \(\text{Card}(A)\).

• Finite set A: cardinality is \(n(A)\)
• Countably infinite set: cardinality is \(\aleph_0\) (aleph-naught)
• Uncountably infinite set: cardinality is \(c\) (continuum)

The concept and notation of cardinality are due to Georg Cantor, who defined the notion and realized that sets can have different cardinalities.

Examples:

• If \(A = \{x : x < 4,\ x \in \mathbb{W}\}\) then \(A = \{0,1,2,3\}\) and \(n(A) = 4\)
• If \(B = \{\text{letters in "mathematics"}\}\) then \(B = \{m,a,t,h,e,i,c,s\}\) and \(n(B) = 8\)

---

Arithmetic of Cardinality

When solving verbal problems involving sets, you perform arithmetic on cardinalities to find unknown values. The three key rules are:

1. Union:   \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\)
2. Complement:   \(n(A') = n(U) - n(A)\)
3. Set Difference:   \(n(A - B) = n(A) - n(A \cap B)\)

---

Set Operation and Cardinality — Reading from Venn Diagrams

Study the given Venn diagrams and find the elements and cardinality of each tabulated set.

SN	Set Notation
1	\(A_o\) (only A)	\(A_o=\{a,b\}\),  \(n_o(A)=2\)	\(A_o=\{1\}\),  \(n_o(A)=1\)	\(A_o=\{1,2,3\}\),  \(n_o(A)=3\)
2	\(B_o\) (only B)
3	\(A \cap B\)
4	\(\overline{A \cup B}\)
5	\(A\)
6	\(\overline{A}\)
7	\(B\)
8	\(\overline{B}\)
9	\(A \triangle B\)
10	\(\overline{A \triangle B}\)
11	\(A \cup B\)
12	\(\overline{A_o}\)
13	\(\overline{B_o}\)
14	\(\overline{A \cap B}\)

---

Cardinality from Labeled Venn Diagrams

When a Venn diagram shows variables instead of elements, use the arithmetic rules to express cardinalities algebraically.

SN	Set Notation
1	\(n_o(A)\) only A	\(n_o(A)=p\)	\(n_o(A)=w\)	\(n_o(A)=a\)