Hello to all our future '68 cohort and students preparing for the TCAS exams!
Welcome to the first lesson of Applied Mathematics 1, which is "Sets". Think of this chapter as the gateway to high school mathematics; it serves as a crucial foundation for Real Numbers, Logic, and Functions. If you build a solid understanding here, the following chapters will become much easier to master!
"If it feels a bit tricky at first, don't worry..." Essentially, sets are just about "grouping" things together. Imagine creating categories for a music playlist or grouping your friends in a classroom—that's exactly what a set is!
1. Getting to Know "Sets" and "Elements"
In mathematics, a Set is a collection of objects that are clearly defined, so we can determine with certainty whether "this object belongs in this group or not."
• We call the objects inside a set Elements.
• We use the symbol \( \in \) to denote membership (e.g., \( a \in A \) is read as "a is an element of set A").
• We use the symbol \( \notin \) to denote that something is not an element.
Important Point: To be considered a set, the criteria must be "well-defined." For example, the "set of vowels in the English alphabet" is a set because everyone knows it includes a, e, i, o, and u. However, a "set of good-looking people in the room" is not considered a set because beauty is subjective and varies from person to person.
2. Ways to Represent a Set (Two methods you need to know)
1) Roster Method (Tabular Form): List every element inside curly brackets \( \{ \dots \} \) separated by commas. For example, the set of positive even numbers less than 10 is \( A = \{2, 4, 6, 8\} \).
2) Set-Builder Notation: Use a variable to represent the elements and describe the rule. For example, \( B = \{ x \mid x \text{ is a day of the week} \} \) (read as: set B consists of all elements x, such that x is a day of the week).
Common Mistakes:
• When writing a set, repeat elements should only be written "once." For example, \( \{1, 1, 2, 3\} \) should be written as \( \{1, 2, 3\} \).
• The order of elements "does not matter." This means \( \{1, 2\} \) is identical to \( \{2, 1\} \).
3. Types of Sets to Know
• Empty Set (Null Set): A set with no elements, represented by \( \emptyset \) or \( \{ \} \). (Remember that \( \{0\} \) is not an empty set because it contains 0 as an element!)
• Finite Set: A set where you can determine the exact number of elements (this includes the empty set, which has 0 elements).
• Infinite Set: A set with an endless number of elements, such as the set of integers \( \{ \dots, -1, 0, 1, \dots \} \).
• Universal Set: The boundary of all objects under consideration, represented by \( U \).
4. Relationships Between Sets: Subsets
A subset is essentially a "smaller collection" within a larger one. If every element of set A is also in set B, we say that A is a subset of B, represented by \( A \subset B \).
A simple analogy: Think of a "pencil case" as being a subset of your "school bag," because everything inside the pencil case is also physically inside the school bag.
Properties to Remember (Frequent Exam Topics!):
1. \( \emptyset \) is a subset of every set.
2. Every set is a subset of itself (\( A \subset A \)).
Power Set: A set that contains "all possible subsets" of a given set, represented by \( P(A) \).
Pro Tip: If set A has \( n \) elements, the number of elements in its Power Set \( P(A) \) is \( 2^n \).
5. Operations on Sets
These are the heart of solving problems. There are 4 main operations:
1) Union (\( \cup \)): "Take everything." Like combining friend groups, \( A \cup B \) includes everything in A, everything in B, or both.
2) Intersection (\( \cap \)): "Take the overlap." \( A \cap B \) includes only the elements that are present in both A and B simultaneously.
3) Difference (\( A - B \)): "Take A, but exclude B." Start with all of A and remove anything that is also found in B.
4) Complement (\( A' \)): "Everything but A." \( A' \) contains every element in the Universal Set (\( U \)) except those belonging to A.
6. Venn Diagrams and Counting Elements
Drawing diagrams helps visualize the relationships clearly. We use rectangles to represent \( U \) and circles for various sets.
The Famous Formula for 2 Sets:
\( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)
Why subtract \( n(A \cap B) \)?
Imagine two overlapping circles. If you count everyone in circle A and then count everyone in circle B, the people in the middle overlap were counted "twice." We must subtract that middle portion once to get the correct total!
Key Takeaways (Must-knows for the exam)
1. Symbols: Distinguish between \( \in \) (is an element of) and \( \subset \) (is a subset of).
2. Empty Set: \( \emptyset \) is a subset of every set, but it is not necessarily an element of every set (check each case carefully).
3. Number of Subsets: \( 2^n \) is the secret key to finding Power Sets.
4. Venn Diagrams: When solving problems, always start by filling in the data from the "most overlapping section" (the center) first, then work your way outwards.
Did you know? The empty set \( \emptyset \) is a set with absolutely nothing in it. However, if you put it inside curly brackets like this: \( \{ \emptyset \} \), it becomes a set with exactly 1 element! (Think of it as an empty box placed inside another box).
How was that? Sets aren't as scary as they seem, right? Try finding some practice problems and drawing Venn diagrams regularly. You’ll soon realize this is one of the most reliable chapters for picking up marks! Keep up the good work!