SETS

A set is a well-defined collection of distinct objects (elements).

1. Notation

   • Sets are denoted by capital letters (e.g., $A$), elements by lowercase (e.g., $a$).

   • Curly braces ${}$ enclose elements.

   • $\in$ means "is an element of," $<br>otin$ means "is not an element of."

   • Example: If $A = {1, 2, 3}$, then $1 \in A$ and $4 \notin A$.

2. Ways to Describe Sets

   • Roster Method: Listing all elements (e.g., $E = {2, 4, 6, 8}$).

     • Example Problem: Express the set of vowels in the English alphabet using the Roster Method.

       • Solution: $V = {a, e, i, o, u}$.

   • Set-Builder Notation: Describing element properties (e.g., $\text{A} = {x \mid x \text{ is an integer and } 1 \le x \le 5}$).

     • Example Problem: Express the set of all positive even integers using Set-Builder Notation.

       • Solution: \text{P} = {x \mid x \text{ is an even integer and } x > 0}. or P = {2n \mid n \in \mathbb{Z}, n > 0}.

3. Special Sets

   • Empty Set ($\emptyset$ or ${}$): Contains no elements.

     • Example Problem: What is the set of all prime numbers divisible by 4?

       • Solution: This is an empty set, written as $\emptyset$ or ${}$ because no prime number is divisible by 4.

   • Universal Set (U): All possible elements under consideration.

   • Singleton Set: Contains exactly one element (e.g., ${ {5} }$).

4. Cardinality of a Set

   • The number of distinct elements in a set, denoted by $\Vert A\Vert$ or $n(A)$.

   • Example: If $A = {a, b, c, d}$, then $\Vert A\Vert = 4$.

   • Example Problem: Find the cardinality of the set $S = {x \mid x \text{ is a day of the week}}$.

     • Solution: $\Vert S\Vert = 7$.

5. Types of Sets by Cardinality

   • Finite Set: Definite, countable number of elements (e.g., days in a week).

   • Infinite Set: Unlimited number of elements (e.g., natural numbers $\mathbb{N}$).

6. Equality of Sets

   • $A = B$ if they contain the same elements, regardless of order or duplicates.

   • Example: ${1, 2, 3} = {3, 1, 2}$.

   • Example Problem: Are the sets $X = {red, blue, green}$ and $Y = {green, red, blue}$ equal?

     • Solution: Yes, $X = Y$ because they contain the exact same elements.

7. Subsets and Supersets

   • Subset ($\subseteq$): Every element of $A$ is in $B$ (e.g., ${1, 2} \subseteq {1, 2, 3}$).

     • Example Problem: Given $A = {a, b}$ and $B = {a, b, c}$, is $A \subseteq B$?

       • Solution: Yes, because all elements of $A$ ($a, b$) are also in $B$.

   • Proper Subset ($\subset$): $A \subseteq B$ but $A \ne B$ (e.g., ${1, 2} \subset {1, 2, 3}$).

     • Example Problem: Using the sets $A = {a, b}$ and $B = {a, b, c}$, is $A \subset B$?

       • Solution: Yes, because $A \subseteq B$ and $B$ contains an element ($c$) not in $A$.

   • Superset ($\supseteq$): If $A \subseteq B$, then $B \supseteq A$.

   • Proper Superset ($\supset$): If $A \subset B$, then $B \supset A$.

8. Set Operations

   • Union ($A \cup B$): Elements in $A$, $B$, or both (e.g., ${1,2,3} \cup {3,4,5} = {1,2,3,4,5}$).

     • Example Problem: If $S<em>1 = {dog, cat}$ and $S</em>2 = {cat, bird}$, find $S<em>1 \cup S</em>2$.

       • Solution: $S<em>1 \cup S</em>2 = {dog, cat, bird}$.

   • Intersection ($A \cap B$): Elements common to both $A$ and $B$ (e.g., ${1,2,3} \cap {3,4,5} = {3}$).

     • Example Problem: If $S<em>1 = {dog, cat}$ and $S</em>2 = {cat, bird}$, find $S<em>1 \cap S</em>2$.

       • Solution: $S<em>1 \cap S</em>2 = {cat}$.

   • Set Difference ($A - B$ or $A \setminus B$): Elements in $A$ but not in $B$ (e.g., ${1,2,3} - {3,4,5} = {1,2}$).

     • Example Problem: If $S<em>1 = {dog, cat}$ and $S</em>2 = {cat, bird}$, find $S<em>1 - S</em>2$.

       • Solution: $S<em>1 - S</em>2 = {dog}$.

   • Complement ($A'$ or $\bar{A}$): All elements in the universal set $U$ that are not in $A$.

     • Example Problem: If the universal set $U = {1, 2, 3, 4, 5}$ and $A = {1, 3, 5}$, find $</p></li></ul></li></ul></li></ol><h5 collapsed="false" seolevelmigrated="true">Experiments</h5><p>An experiment is a process that results in an outcome that cannot be predicted with certainty. It is a procedure that can be repeated under the same conditions, and its outcome is one of several possible results.</p><h5 collapsed="false" seolevelmigrated="true">Sample Spaces</h5><h6 collapsed="false" seolevelmigrated="true">Definition</h6><p>The sample space, denoted by$S$or$\Omega$, is the set of all possible outcomes of a random experiment.</p><h6 collapsed="false" seolevelmigrated="true">Properties</h6><ul><li><p>Each outcome in the sample space is called a sample point or element.</p></li><li><p>The sample space can be finite or infinite.</p></li></ul><h6 collapsed="false" seolevelmigrated="true">Examples</h6><ul><li><p><strong>Tossing a coin once</strong>:$S = {\text{Head, Tail}}$</p></li><li><p><strong>Rolling a six-sided die once</strong>:$S = {1, 2, 3, 4, 5, 6}$</p></li><li><p><strong>Tossing two coins</strong>:$S = {\text{(H,H), (H,T), (T,H), (T,T)}}$</p></li></ul><h5 collapsed="false" seolevelmigrated="true">Events</h5><h6 collapsed="false" seolevelmigrated="true">Definition</h6><p>An event, denoted by a capital letter (e.g.,$A$,$B$,$E$), is any subset of the sample space. It is a collection of specific outcomes of an experiment.</p><h6 collapsed="false" seolevelmigrated="true">Types of Events</h6><ul><li><p><strong>Simple Event</strong>: An event containing exactly one outcome (e.g., rolling a 3 on a die,$E = {3}$).</p></li><li><p><strong>Compound Event</strong>: An event containing more than one outcome (e.g., rolling an even number on a die,$E = {2, 4, 6}$).</p></li><li><p><strong>Impossible Event</strong>: An event that contains no outcomes (i.e., the empty set$\emptyset$).</p></li><li><p><strong>Certain Event</strong>: An event that contains all outcomes in the sample space (i.e., the sample space$S$itself).</p></li></ul><h6 collapsed="false" seolevelmigrated="true">Examples</h6><ul><li><p><strong>Rolling a six-sided die</strong>: Let$S = {1, 2, 3, 4, 5, 6}$</p><ul><li><p>Event$A$: Rolling an even number.$A = {2, 4, 6}$</p></li><li><p>Event$B$: Rolling a number greater than 4.$B = {5, 6}$</p></li><li><p>Event$C$: Rolling a 1.$C = {1}$</p></li></ul></li></ul><p>A set is a well-defined collection of distinct objects (elements). 1. Notation - Sets are denoted by capital letters (e.g.,$A$), elements by lowercase (e.g.,$a$). - Curly braces${}$enclose elements. -$\in$means "is an element of,"$\notin$means "is not an element of." - Example: If$A = {1, 2, 3}$, then$1 \in A$and$4 \notin A$. 2. Ways to Describe Sets - **Roster Method:** Listing all elements (e.g.,$E = {2, 4, 6, 8}$). - **Example Problem:** Express the set of vowels in the English alphabet using the Roster Method. - **Solution:**$V = {a, e, i, o, u}$. - **Set-Builder Notation:** Describing element properties (e.g.,$\text{A} = {x \mid x \text{ is an integer and } 1 \le x \le 5}$). - **Example Problem:** Express the set of all positive even integers using Set-Builder Notation. - **Solution:**$\text{P} = {x \mid x \text{ is an even integer and } x > 0}$. or$P = {2n \mid n \in \mathbb{Z}, n > 0}$. 3. Special Sets - **Empty Set ($\emptyset$or${}$):** Contains no elements. - **Example Problem:** What is the set of all prime numbers divisible by 4? - **Solution:** This is an empty set, written as$\emptyset$or${}$because no prime number is divisible by 4. - **Universal Set (U):** All possible elements under consideration. - **Singleton Set:** Contains exactly one element (e.g.,$ { {5} } $). 4. Cardinality of a Set - The number of distinct elements in a set, denoted by$\Vert A\Vert$or$n(A)$. - Example: If$A = {a, b, c, d}$, then$\Vert A\Vert = 4$. - *!Example Problem:*! Find the cardinality of the set$S = {x \mid x \text{ is a day of the week}}$. - **Solution:**$\Vert S\Vert = 7$. 5. Types of Sets by Cardinality - **Finite Set:** Definite, countable number of elements (e.g., days in a week). - **Infinite Set:** Unlimited number of elements (e.g., natural numbers$\mathbb{N}$). 6. Equality of Sets -$A = B$if they contain the same elements, regardless of order or duplicates. - Example:${1, 2, 3} = {3, 1, 2}$. - **Example Problem:** Are the sets$X = {red, blue, green}$and$Y = {green, red, blue}$equal? - **Solution:** Yes,$X = Y$because they contain the exact same elements. 7. Subsets and Supersets - **Subset ($\subseteq$):** Every element of$A$is in$B$(e.g.,${1, 2} \subseteq {1, 2, 3}$). - **Example Problem:** Given$A = {a, b}$and$B = {a, b, c}$, is$A \subseteq B$? - **Solution:** Yes, because all elements of$A$($a, b$) are also in$B$. - **Proper Subset ($\subset$):**$A \subseteq B$but$A \ne B$(e.g.,${1, 2} \subset {1, 2, 3}$). - **Example Problem:** Using the sets$A = {a, b}$and$B = {a, b, c}$, is$A \subset B$? - **Solution:** Yes, because$A \subseteq B$and$B$contains an element ($c$) not in$A$. - **Superset ($\supseteq$):** If$A \subseteq B$, then$B \supseteq A$. - **Proper Superset ($$\supset$):** If$A \subset B$, then$B \supset A$. 8. Set Operations - **Union ($A \cup B$):** Elements in$A$,$B$, or both (e.g.,${1,2,3} \cup {3,4,5} = {1,2,3,4,5}$). - **Example Problem:** If$S1 = {dog, cat}$and$S2 = {cat, bird}$, find$S1 \cup S2$. - **Solution:**$S1 \cup S2 = {dog, cat, bird}$. - **Intersection ($A \cap B$):** Elements common to both$A$and$B$(e.g.,${1,2,3} \cap {3,4,5} = {3}$). - **Example Problem:** If$S1 = {dog, cat}$and$S2 = {cat, bird}$, find$S1 \cap S2$. - **Solution:**$S1 \cap S2 = {cat}$. - **Set Difference ($A - B$or$A \setminus B$):** Elements in$A$but not in$B$(e.g.,${1,2,3} - {3,4,5} = {1,2}$). - **Example Problem:** If$S1 = {dog, cat}$and$S2 = {cat, bird}$, find$S1 - S2$. - **Solution:**$S1 - S2 = {dog}$. - **Complement ($A'$or$\bar{A}$):** All elements in the universal set$U$that are not in$A$. - **Example Problem:** If the universal set$U = {1, 2, 3, 4, 5}$and$A = {1, 3, 5}$, find$A'$. - **Solution:**$A' = {2, 4}. ### Experiments
       An experiment is a process that results in an outcome that cannot be predicted with certainty. It is a procedure that can be repeated under the same conditions, and its outcome is one of several possible results. ### Sample Spaces

       Definition

       The sample space, denoted by S$or$\Omega, is the set of all possible outcomes of a random experiment. ##### Properties

       • Each outcome in the sample space is called a sample point or element.

       • The sample space can be finite or infinite. ##### Examples

       • **Tossing a coin once**: S = {\text{Head, Tail}}$</p></li><li><p>**Rolling a six-sided die once**:$S = {1, 2, 3, 4, 5, 6}$</p></li><li><p>**Tossing two coins**:$S = {\text{(H,H), (H,T), (T,H), (T,T)}} ### Events

       Definition

       An event, denoted by a capital letter (e.g., A$,$B$,$E), is any subset of the sample space. It is a collection of specific outcomes of an experiment. ##### Types of Events

       • **Simple Event**: An event containing exactly one outcome (e.g., rolling a 3 on a die, E = {3}$).</p></li><li><p>**Compound Event**: An event containing more than one outcome (e.g., rolling an even number on a die,$E = {2, 4, 6}$).</p></li><li><p>**Impossible Event**: An event that contains no outcomes (i.e., the empty set$\emptyset$).</p></li><li><p>**Certain Event**: An event that contains all outcomes in the sample space (i.e., the sample space$S itself). ##### Examples

       • **Rolling a six-sided die**: Let S = {1, 2, 3, 4, 5, 6}$- Event$A$: Rolling an even number.$A = {2, 4, 6}$- Event$B$: Rolling a number greater than 4.$B = {5, 6}$- Event$C$: Rolling a 1.$C = {1} ### Probability

       Definition

       Probability is a numerical measure of the likelihood of an event occurring. It is a value between 0 and 1, inclusive, where 0 indicates impossibility and 1 indicates certainty.

       Classical Probability

       For an event E$in a sample space$S$with equally likely outcomes, the probability of$E$is given by the formula:<br>$P(E) = \frac{\text{Number of favorable outcomes (in E)}}{\text{Total number of possible outcomes (in S)}}$</p><h6 collapsed="false" seolevelmigrated="true">Properties</h6><ul><li><p>The probability of any event$E$is between 0 and 1:$0 \le P(E) \le 1$</p></li><li><p>The sum of probabilities of all possible outcomes in a sample space is 1.</p></li><li><p>The probability of an impossible event is 0:$P(\emptyset) = 0$</p></li><li><p>The probability of a certain event (the sample space$S$) is 1:$P(S) = 1$</p></li></ul><h6 collapsed="false" seolevelmigrated="true">Example Problem</h6><p>What is the probability of rolling an even number when a fair six-sided die is rolled once?</p><h6 collapsed="false" seolevelmigrated="true">Solution</h6><ul><li><p>The sample space is$S = {1, 2, 3, 4, 5, 6}$, so the total number of outcomes is$\Vert S\Vert = 6$.</p></li><li><p>The event of rolling an even number is$E = {2, 4, 6}$, so the number of favorable outcomes is$\Vert E\Vert = 3$.</p></li><li><p>The probability is$P(E) = \frac{3}{6} = \frac{1}{2} = 0.5$$.