Set Theory Basics: Elements, Cardinality, and Set Equality

Set Theory Fundamentals: Element Count and Set Equality

Question 3.a: Is 4 = {4}?
- Answer: No, 4 \neq {4}.
- Explanation: The number 4 is an integer, a single numerical value. The expression {4} represents a set that contains the integer 4 as its only element. An element is not the same as a set containing that element. Think of 4 as a single apple, and {4} as a box containing exactly one apple. The apple is not the same as the box.

Principle: When counting elements in a set, only distinct elements are considered. Duplicate entries do not increase the count.
Question 3.b: How many elements are in the set {3, 4, 3, 5}?
- Answer: There are 3 elements.
- Explanation: The set {3, 4, 3, 5} is equivalent to {3, 4, 5}. The element 3 is listed twice, but it is still only one distinct element. The unique elements are 3, 4, and 5.
Question 3.c: How many elements are in the set {1, {1}, {1, {1}}}
- Answer: There are 3 elements.
- Explanation: This set contains three distinct elements:
  1. The integer 1
  2. The set {1} (which is distinct from the integer 1)
  3. The set {1, {1}} (which is distinct from both the integer 1 and the set {1})
    Each of these is a unique, distinguishable entity within the larger set.

Question 4.a: 2 \in {2}?
- Answer: Yes, 2 \in {2}. (The symbol \in means "is an element of" or "is a member of").
- Explanation: The statement asks if the integer 2 is an element of the set {2}. By definition, the set {2} contains the integer 2 as its only element.
Question 4.b: How many elements are in the set {2, 2, 2, 2}?
- Answer: There is 1 element.
- Explanation: Similar to 3.b, duplicates are not counted. The set {2, 2, 2, 2} is equivalent to {2} because it contains only one unique element, the integer 2.
Question 4.c: How many elements are in the set {0, {0}}
- Answer: There are 2 elements.
- Explanation: This set contains two distinct elements:
  1. The integer 0
  2. The set {0} (which is distinct from the integer 0)
Question 4.d: Is {0} \in {{0}, {1}}
- Answer: Yes, {0} \in {{0}, {1}}.
- Explanation: The set {{0}, {1}} explicitly lists two elements: the set {0} and the set {1}. Since {0} is one of these listed elements, it is a member of the larger set.
Question 4.e: Is 0 = {{0}, {1}}
- Answer: No, 0 \neq {{0}, {1}}.
- Explanation: The integer 0 is a number. The expression {{0}, {1}} is a set containing two other sets. An integer can never be equal to a set, conceptually or mathematically.

Principle: Two sets are equal if and only if they contain exactly the same elements. The order of elements does not matter, nor does the repetition of elements.
Let's analyze each given set:
- Set A: A = {0, 1, 2} (This set lists specific integers directly.)
- Set B: B = {x \in \mathbb{R} | -1 \le x < 3} (This is the set of all real numbers (\mathbb{R}) such that x is greater than or equal to -1 and strictly less than 3).
  - In interval notation, this is [-1, 3). It contains an infinite number of real numbers, including fractions and irrational numbers, between -1 and 3.
- Set C: C = {x \in \mathbb{R} | -1 < x < 3} (This is the set of all real numbers (\mathbb{R}) such that x is strictly greater than -1 and strictly less than 3).
  - In interval notation, this is (-1, 3). It also contains an infinite number of real numbers, excluding -1 itself.
- Set D: D = {x \in \mathbb{Z} | -1 < x < 3} (This is the set of all integers (\mathbb{Z}) such that x is strictly greater than -1 and strictly less than 3).
  - The integers that satisfy this condition are 0, 1, 2. So, D = {0, 1, 2}.
- Set E: E = {x \in \mathbb{Z} | -1 < x < 3} (This set has the exact same definition as Set D).
  - Therefore, E = {0, 1, 2}.

Comparing the explicit elements of each set:
- A = {0, 1, 2} (Contains three specific integers)
- B = [-1, 3) (Contains an infinite number of real numbers)
- C = (-1, 3) (Contains an infinite number of real numbers)
- D = {0, 1, 2} (Contains three specific integers)
- E = {0, 1, 2} (Contains three specific integers)
Question 5: Which of the following sets are equal?
- Answer: Sets A, D, and E are equal.
- Explanation: Sets A, D, and E all contain precisely the same elements: the integers 0, 1, and 2. Sets B and C are fundamentally different because they comprise real numbers within an interval, which means they contain an infinite number of elements, including non-integers, unlike sets A, D, and E which only contain three specific integers.