Lecture Notes: Logic, Functions, and Inequalities

Conditional Statements and Truth Values

• Notation: we read this as $p \implies q$ (p implies q).
• Truth condition: the conditional is true in all cases except when the hypothesis (p) is true and the conclusion (q) is false. In that case, the statement is false; otherwise it is true.
• Intuition: the statement is often summarized as “it is true unless p is true and q is false.”
• Converse, contrapositive, and their relevance:
  • Converse: $q \implies p$ (swapping hypothesis and conclusion).
  • Contrapositive: $\neg q \implies \neg p$ (negating both parts and swapping them).
  • The contrapositive preserves the meaning of the original conditional; if $p \implies q$ is true, then its contrapositive $\neg q \implies \neg p$ is also true.
  • Notation: equivalence of statements is used to express that two statements have the same truth value for all possible inputs. This is denoted by symbols like $\equiv$ or by the phrase "is equivalent to".
• Logical equivalence:
  • A statement $p \implies q$ is equivalent to its contrapositive $\neg q \implies \neg p$.
  • The three-line notation used in class sometimes denotes logical equivalence with a triple bar: $\;\equiv\;$ or a specialized symbol for "is equivalent to".
  • In many contexts, equivalence is written with a double-arrow relation (e.g., $p \iff q$) to mean both $p \implies q$ and $q \implies p$ hold.

Functions and Mappings

• Function definition (precision): A function is a mapping from a domain to a range such that every input in the domain is mapped to exactly one output in the range. Formally, if the domain is $D$ and the range is $R$, then a function is a rule $f: D \to R$ with for all $x \in D$, there exists a unique $y \in R$ such that $f(x) = y$.
• Intuition: a function assigns each input a single output (consistency).
• Example:
  • The function $f(x) = x^2$ is a mapping from the real numbers to the real numbers; for example, $f(2) = 4$ and $f(-2) = 4$, so different inputs can share the same output.
• Notation: when we input an element $x \in D$, the function outputs $f(x)$.
• Important distinction: a function can map multiple inputs to the same output, but it cannot give two outputs for a single input.
• Connection to the conditional view:
  • A function can be viewed as a process with input (domain) and output (range), where the mapping acts like a rule: under certain inputs we obtain certain outputs (akin to a hypothesis → conclusion structure).
• Quick example: the mapping with growth or decay, or a simple polynomial like $f(x) = x^2$ demonstrates that two different inputs may yield the same output (e.g., $x=2$ and $x=-2$ both give $f(x)=4$).

Negation of Statements

• Negation: given a statement $p$, its negation is $\neg p$, meaning the opposite truth value of $p$.
• Notation examples:
  • If "The ground is wet" is true, then the negation "The ground is not wet" is false, and vice versa.
• Practical note: negation is straightforward for simple statements, but becomes more complex for longer sentences with connectives (AND, OR).

The Contrapositive and Logical Equivalence

• Contrapositive: for a conditional $p \implies q$, the contrapositive is $\neg q \implies \neg p$.
• Truth preservation: the contrapositive has the same truth value as the original conditional; they are logically equivalent.
• Example: If it barks, it is a dog (true/false depending on day-to-day world), its contrapositive is: If it is not a dog, it does not bark.
• Logical equivalence definition: two statements are logically equivalent if they have the same truth value for all possible inputs. It is denoted as $p \implies q \equiv \neg q \implies \neg p$ in many texts, and often written with a triple bar $\equiv$ or with a double arrow $\iff$ to indicate mutual implication.
• Practical takeaway: logical equivalence allows solving problems from a different perspective; if one form is easier to reason about, the other is equally valid.

And / Or Logic

• AND (conjunction): $p \land q$ is true only when both $p$ and $q$ are true.
• OR (disjunction): $p \lor q$ is true when at least one of $p$ or $q$ is true. If both are true, $p \lor q$ is also true.
• Note on exclusive OR: exclusive OR ("one or the other, but not both") is a different operator and not the focus here.

Connecting Algebra to Logic: Undos, Implication Errors, and Proofs

• When solving equations like $x^2 = 4$, naive reasoning might lead to only $x = 2$. Truth is, both $x = 2$ and $x = -2$ satisfy the equation.
• Proof approach for conditionals:
  • To prove $p \implies q$, assume the hypothesis $p$ is true and then deduce whether the conclusion $q$ must be true.
  • If you can derive $q$ from the assumption, the implication is proven (provided you don’t rely on unjustified leaps).
  • Show your logical steps clearly (the lecture emphasizes writing down the deduction steps from hypothesis to conclusion).
• Important example: solving $x^2 = 4$:
  • Assume $x^2 = 4$; take square roots (consider ±): you get $x = 2$ or $x = -2$.
  • Therefore the conclusion is true for both possible solutions; the original conditional statement about the roots is considered true in the sense of solving the equation fully.
• The instructor warns against skipping steps: always show enough logical steps to justify each implication in the deduction.

Absolute Values and Logical Decomposition

• Key idea: absolute value allows a natural split into two cases (or a single case with an AND/OR combination depending on the inequality sign).
• General rules:
  • If |A| > a (with a>0), then A > a \quad\text{or}\quad A < -a.
  • If $|A| < a$ (with $a>0$), then -a < A < a (i.e., two inequalities joined by AND).
• This translates algebraically to splitting into two intervals on the number line and taking intersections for AND cases.
• Example: solving |x| > 3 gives x > 3 or x < -3.
• Example: solving |x| < 3 gives -3 < x < 3 (the AND of two inequalities).
• Interval notation intuition:
  • |x| > 3 corresponds to the union of two rays: $(-\infty,-3) \cup (3,\infty)$.
  • |x| < 3 corresponds to the interval: $(-3,3)$.

Example Problem: Inequality Involving Absolute Value

• Problem: Find all values of $x$ that satisfy
  $-3|x+2| + 5 \geq -2.$
• Step 1: Isolate the absolute value
  • Subtract 5 from both sides: $-3|x+2| \geq -7$
  • Divide both sides by $-3$ (note the inequality flips): $|x+2| \leq \frac{7}{3}$
• Step 2: Split into the two-sided inequality for the absolute value
  • This is equivalent to
    $-\frac{7}{3} \leq x+2 \leq \frac{7}{3}$
• Step 3: Solve for $x$ by subtracting 2 throughout
  • Subtract 2: $-\frac{13}{3} \leq x \leq \frac{1}{3}$
• Step 4: Conclude in interval notation
  • The solution set is $[-\frac{13}{3}, \frac{1}{3}]$
• Discussion: the decomposition used here mirrors AND/OR logic: the absolute value bounds translate into a pair of inequalities joined by AND, or into a pair of inequalities joined by OR depending on the direction of the inequality.
• Practical tip: when you see |x| > a, think OR (two separate intervals). When you see |x| < a, think AND (the intersection of two inequalities).

Practice and Exam Strategy

• Don’t just memorize truth tables; understand the logic structure and how to rewrite statements using negation, contrapositive, and equivalence.
• On exams, you may be asked to reorder or rewrite a conditional (e.g., find the contrapositive or show the equivalence). The underlying truth values remain the same.
• When solving equations or inequalities that involve conditional statements or absolute values, clearly separate cases and show each logical step, especially when applying inverse operations or taking roots.
• Expect problems that combine algebra with logic (e.g., using AND/OR to express absolute value inequalities) and be comfortable translating between the algebraic and logical forms.

Quick Recap of Symbols and Key Facts

• Conditional: $p \implies q$; true except when $p$ is true and $q$ is false.
• Negation: $\neg p$; opposite truth value of $p$.
• Contrapositive: $\neg q \implies \neg p$; logically equivalent to the original statement.
• Equivalence: two statements have the same truth value for all inputs; often written as $p \iff q$ or $p \implies q \equiv \neg q \implies \neg p$.
• AND: $p \land q$ is true only if both are true.
• OR: $p \lor q$ is true if at least one is true (inclusive OR).
• Absolute value rules:
  • |A| > a \iff A > a \text{ or } A < -a
  • |A| < a \iff -a < A < a
• Example solution to a concrete inequality: for
  $-3|x+2| + 5 \geq -2$
  the solution is $x \in [-\frac{13}{3}, \frac{1}{3}]$.