math final

module 1: logic

1. statements and negations

• statement: a declarative sentence that can be objectively judged as either true or false, but not both. It expresses a complete thought. Opinions (like "James is tall") are not statements because they are subjective and lack objective truth value.

  • Example: "The sky is blue" (can be true or false depending on the time of day/weather, but it's a statement).

  • Example: "$2 + 2 = 5$ " (a false statement).

• negation (~): a statement that has the opposite truth value of the original statement. If the original statement is true, its negation is false, and vice versa.

  • Symbol: ~p means "not p".

  • Example: If p is "It is raining," then ~p is "It is not raining."

2. logical connectives

• conjunction (∧): read as "and". This compound statement is true only if both of its component parts (simple statements) are true. If even one part is false, the entire conjunction is false.

  • Truth Table for p ∧ q:

  • p is T, q is T → p ∧ q is T

  • p is T, q is F → p ∧ q is F

  • p is F, q is T → p ∧ q is F

  • p is F, q is F → p ∧ q is F

• disjunction (∨): read as "or". This compound statement is true if at least one of its component parts is true (inclusive OR). It is only false if both component parts are false.

  • Truth Table for p ∨ q:

  • p is T, q is T → p ∨ q is T

  • p is T, q is F → p ∨ q is T

  • p is F, q is T → p ∨ q is T

  • p is F, q is F → p ∨ q is F

• conditional (→): read as "if… then". Expressed as "If p, then q," where p is the hypothesis (antecedent) and q is the conclusion (consequent).

  • The "promise" rule: Think of the statement like a promise (e.g., "If I bring pie, then I will bring ice cream"). The statement is false only when the promise is broken: you did the first part (brought pie, p is T) but failed the second part (no ice cream, q is F). In all other cases, the conditional statement is considered true.

  • Truth Table for p → q:

  • p is T, q is T → p → q is T (Promise kept)

  • p is T, q is F → p → q is F (Promise broken)

  • p is F, q is T → p → q is T (No pie, but still brought ice cream - promise not broken)

  • p is F, q is F → p → q is T (No pie, no ice cream - promise not broken)

• biconditional (↔): read as "if and only if" (often abbreviated as "iff"). This compound statement is true when both component parts have the same truth value (both true or both false). It implies that (p → q) ∧ (q → p).

  • Truth Table for p ↔ q:

  • p is T, q is T → p ↔ q is T

  • p is T, q is F → p ↔ q is F

  • p is F, q is T → p ↔ q is F

  • p is F, q is F → p ↔ q is T

3. truth tables

• conjunction (and): true only when p is T and q is T.

• disjunction (or): false only when p is F and q is F.

• conditional (if/then): false only when the hypothesis is true and the conclusion is false (T → F).

• biconditional: true when p and q have the same truth value (T ↔ T or F ↔ F).

module 2: sets

1. set notation

• set: a well-defined collection of distinct objects, called elements, where the order of elements does not matter. "Well-defined" means it is possible to determine whether any given object belongs to the set or not.

  • Example: The set of even numbers less than 10 is {2, 4, 6, 8}.

• element (∈): an object belonging to a set.

  • Symbol: x ∈ A means "x is an element of set A."

  • Symbol: x ∉ A means "x is not an element of set A."

• notation types:

  • roster: lists all the elements of the set, typically enclosed in curly braces {}.

  • Example: {1, 2, 3} or {red, blue, green}.

  • set-builder: describes the properties that elements of the set must satisfy.

  • Example: {\text{x \mid x is a natural number and x < 5}} which reads "the set of all x such that x is a natural number and x is less than 5."

• standard sets:

  • N: natural numbers {1, 2, 3...} (positive integers, used for counting).

  • W: whole numbers {0, 1, 2...} (natural numbers plus zero).

  • I (or Z): integers {... -2, -1, 0, 1, 2 ...} (whole numbers and their negatives).

  • R: real numbers (all numbers that can be represented on a continuous number line, including rational and irrational numbers).

  • Q: rational numbers (numbers that can be expressed as a fraction p/q where p, q ∈ I and q ≠ 0).

2. subsets

• universal set (U): the set of all elements being considered in a particular context. All other sets in that context are subsets of the universal set.

• empty set (∅ or {}): the unique set containing no elements. It is a subset of every set.

  • Note: {0} is not the empty set; it is a set containing one element, which is the number zero.

• subset (⊆): Set B is a subset of set A (written B ⊆ A) if every element in B is also in A. It is possible for B to be equal to A.

  • Example: If A = {1, 2, 3} and B = {1, 2}, then B ⊆ A.

  • Example: {1, 2, 3} ⊆ {1, 2, 3}.

• proper subset (⊂): Set B is a proper subset of set A (written B ⊂ A) if B is a subset of A, but B is not equal to A. This means A must contain at least one element not found in B.

  • Example: If A = {1, 2, 3} and B = {1, 2}, then B ⊂ A.

3. set operations & venn diagrams

• complement (A'): the set of all elements in the universal set U that are not in set A. It is defined as A' = {\text{x \mid x \in U and x \notin A}}.

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• intersection (∩): the set of all elements common to both set A and set B (the overlap between the sets). It is defined as A \cap B = {\text{x \mid x \in A and x \in B}}.

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• union (∪): the set of all elements that belong to set A or set B (or both). It includes everything in both circles of a Venn diagram. It is defined as A \cup B = {\text{x \mid x \in A or x \in B}}.

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• difference (A - B): the set of all elements in set A that are not in set B. It is defined as A - B = {\text{x \mid x \in A and x \notin B}}.

• cartesian product (A × B): the set of all possible ordered pairs (a, b) where the first element a is from set A and the second element b is from set B.

  • Example: if A={1,2} and B={a,b}, then A×B = {(1,a), (1,b), (2,a), (2,b)}.

  • The cardinality (number of elements) of the Cartesian product is $|A \times B| = |A| \cdot |B|$.

module 3: real numbers

1. interval notation & inequalities

• inequalities: used to describe a range of values for a variable.

  • and: use parentheses ( ) in interval notation to indicate that the endpoint is not included (open interval).

  • ≤ or ≥: use brackets [ ] in interval notation to indicate that the endpoint is included (closed interval).

  • Infinity (∞) and negative infinity (-∞) always use parentheses, as they are not true numbers that can be included.

  • Example: −5 < x ≤ −1 becomes $(-5, -1]$.

  • Example: $x ≥ 3$ becomes $[3, \infty)$.

  • Example: x < 7 becomes $(- \infty, 7)$.

2. integer operations

• addition:

  • Same signs = add the absolute values of the numbers and keep the common sign.

  • Different signs = subtract the smaller absolute value from the larger absolute value, and keep the sign of the number with the larger absolute value.

• subtraction: "add the opposite" (keep-change-change). Change the subtraction sign to addition and change the sign of the second number. Then follow the rules for addition.

  • Example: $5 - (-3) = 5 + 3 = 8$.

  • Example: $5 - 7 = 5 + (-7) = -2$.

• multiplication/division:

  • Same signs = positive answer.

  • Different signs = negative answer.

  • Example: $(-4) \cdot (-3) = 12$.

  • Example: $(-10) \div 2 = -5$.

3. calculator cheat codes (ti-30xs multiview)

• setup: set your mode to mathprint so fractions look like real fractions.

  • Press mode → arrow down to bottom → arrow right to MATHPRINT → enter.

• fractions to decimals:

  • Type fraction → press 2nd then f↔d (table button) → enter.

• decimals to fractions:

  • Type decimal → press 2nd then f↔d → enter.

• mixed numbers to improper fractions:

  • Press 2nd then U n/d (above the fraction key) → enter the whole number, numerator, and denominator.

• simplify fractions:

  • Just typing the fraction and hitting enter will automatically reduce it to lowest terms.

4. solving equations

• golden rule: whatever you do to one side of the equation, you must do to the other side to maintain equality.

• reverse order of operations: to isolate the variable, you "undo" operations in the reverse order of PEMDAS (Parentheses, Exponents, Multiplication & Division, Addition & Subtraction).

  • Step 1: Undo addition/subtraction (by performing the inverse operation on both sides).

  • Step 2: Undo multiplication/division (by performing the inverse operation on both sides).

  • Example: To solve $2x + 5 = 11$

  • Subtract 5 from both sides: $2x = 6$.

  • Divide by 2 on both sides: $x = 3$.

5. percents

• percent: means "per 100" or "out of 100". It's a way to express a part of a whole as a fraction of 100.

• conversions:

  • Percent to decimal: divide by 100 (move decimal 2 places left).

  • Example: $25\% = 0.25$.

  • Decimal to percent: multiply by 100 (move decimal 2 places right).

  • Example: $0.75 = 75\%$.

• percent equation: use the proportion $\frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100}$ or the equation $\text{percent} \cdot \text{whole} = \text{part}$ (where percent is in decimal form).

  • Example: What is $20\%$ of $50$?

  • Using the equation: $0.20 \cdot 50 = 10$.

  • Using the proportion: $\frac{\text{part}}{50} = \frac{20}{100} \Rightarrow \text{part} = 50 \cdot \frac{20}{100} = 10$.