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Set Theory and Logic

In mathematics, a Set is just a collection of distinct objects, considered as an object in its own right. Sets are usually denoted by capital letters like (A) or (B), and their contents are written inside curly braces ({}). For example, the set of even numbers from 1 to 5 could be written as (A = {2, 4}).

Element Of

When we want to say that a specific number is inside a set, we use the "element of" symbol: (\in).

  • \(2 \in A\) is read as "2 is an element of set A".
  • To say something is not in the set, we cross it out: (\notin). So, \(3 \notin A\) means "3 is not an element of set A".

Subsets

Sometimes one set is entirely contained within another. For example, if (B = {1, 2, 3, 4, 5}), then every number in set (A) is also in set (B). We say that (A) is a subset of (B). The symbol for subset is (\subset) or (\subseteq).

  • (A \subset B) means "(A) is a subset of (B)". Think of it like a "less than or equal to" sign, but for sets.

Union and Intersection

When working with multiple sets, you often want to combine them or find what they have in common.

  • Union ((\cup)): This looks like a large "U" and means "combine everything". (A \cup B) includes all elements that are in (A), or in (B), or in both.
  • Intersection ((\cap)): This looks like an upside-down "U" and means "only what they share". (A \cap B) includes only the elements that are present in both (A) and (B).

Mini-Quiz

Question 1

How do you read the mathematical statement (x \in S)?

  1. (x) is equal to (S)
  2. (x) is an element of set (S)
  3. (x) is not included in (S)
  4. (x) is a subset of (S)

Correct Answer: B Explanation: The symbol (\in) means "is an element of" or "belongs to".

Question 2

What does the intersection symbol ((\cap)) represent when used between two sets, like (A \cap B)?

  1. All items that are in either set A or set B
  2. Only the items that are present in both set A and set B
  3. Items in set A that are not in set B
  4. The sum of the number of items in both sets

Correct Answer: B Explanation: The intersection ((\cap)) of two sets only includes the elements they share in common.

Question 3

If every asset in Portfolio (X) is also contained within the larger Market Index (Y), how could you write this relationship using set theory symbols?

  1. (X \in Y)
  2. (X \cup Y)
  3. (X \subset Y)
  4. \(X ≈ Y\)

Correct Answer: C Explanation: The symbol (\subset) means "is a subset of". Since all of (X)'s assets are in (Y), (X) is a subset of (Y).