### Key Concepts

### 2.1 Statements and Quantifiers

- Logical statements have the form of a complete sentence and make claims that can be identified as true or false.
- Logical statements are represented symbolically using a lowercase letter.
- The negation of a logical statement has the opposite truth value of the original statement.
- Be able to
- Determine whether a sentence represents a logical statement.
- Write and translate logical statements between words and symbols.
- Negate logical statements, including logical statements containing quantifiers of
*all, some, and none*.

### 2.2 Compound Statements

- Logical connectives are used to form compound logical statements by using words such as
*and, or, and if …, then*. - A conjunction is a compound logical statement formed by combining two statements with the words “and” or “but.” If the two independent clauses are represented by $p$ and $q$, respectively, then the conjunction is written symbolically as $p\wedge q$. For the conjunction to be true, both $p$ and $q$ must be true.
- A disjunction joins two logical statements with the
*or*connective. In, logic*or*is inclusive. For an*or*statement to be true at least one statement must be true, but both may also be true. - A conditional statement has the form if $p$, then $q$, where $p$ and $q$ are logical statements. The only time the conditional statement is false is when $p$ is true, and $q$ is false.
- The biconditional statement is formed using the connective $\mathrm{if\; and\; only\; if}$ for the biconditional statement to be true, the true values of $p$ and $q$, must match. If $p$ is true then $q$ must be true, if $p$ is false, then $q$ must be false.
- Translate compound statements between words and symbolic form.
Connective Symbol Name and

but$\wedge $ conjunction or $\vee $ disjunction, inclusive or not ~ negation if $\dots $, then implies $\to $ conditional, implication if and only if $\leftrightarrow $ biconditional - The dominance of connectives explains the order in which compound logical statements containing multiple connectives should be interpreted.
- The dominance of connectives should be applied in the following order
- Parentheses
- Negations
- Disjunctions/Conjunctions, left to right
- Conditionals
- Biconditionals

### 2.3 Constructing Truth Tables

- Determine the true values of logical statements involving negations, conjunctions, and disjunctions.
- The negation of a logical statement has the opposite true value of the original statement.
- A conjunction is true when both $p$ and $q$ are true, otherwise it is false.
- A disjunction is false when both $p$ and $q$ are false, otherwise it is true.

- Know how to construct a truth table involving negations, conjunctions, and disjunctions and apply the dominance of connectives to determine the truth value of a compound logical statement containing, negations, conjunctions, and disjunctions.
Negation Conjunction (AND) Disjunction (OR) $p$ $~p$ $p$ $q$ $p\wedge q$ $p$ $q$ $p\vee q$ T F T T T T T T F T T F F T F T F T F F T T F F F F F F - A logical statement is valid if it is always true. Know how to construct a truth table for a compound statement and use it to determine the validity of compound statements involving negations, conjunctions, and disjunctions.

### 2.4 Truth Tables for the Conditional and Biconditional

- The conditional statement, if $p$ then $q$, is like a contract. The only time it is false is when the contract has been broken. That is, when $p$ is true, and $q$ is false.
Conditional $p$ $q$ $p\to q$ T T **T**T F **F**F T **T**F F **T** - The biconditional statement, $p$ if and only if $q$, it true whenever $p$ and $q$ have matching true values, otherwise it is false.
Biconditional $p$ $q$ $p\leftrightarrow q$ T T **T**T F **F**F T **F**F F **T** - Know how to construct truth tables involving conditional and biconditional statements.
- Use truth tables to analyze conditional and biconditional statements and determine their validity.

### 2.5 Equivalent Statements

- Two statements $p$ and $q$ are logically equivalent if the biconditional statement, $p\leftrightarrow q$ is a valid argument. That is, the last column of the truth table consists of only true values. In other words, $p\leftrightarrow q$ is a tautology. Symbolically, $p$ is logically equivalent to $q$ is written as: $p\equiv q.$
- A logical statement is a tautology if it is always true.
- To be valid a local argument must be a tautology. It must always be true.
- Know the variations of the conditional statement, be able to determine their truth values and compose statements with them.

- The converse of a conditional statement, if $p$ then $q$, is the statement formed by interchanging the hypothesis and conclusion. It is the statement if $q$ then $p$.
- The inverse of a conditional statement if formed by negating the hypothesis and the conclusion of the conditional statement.
- The contrapositive negates and interchanges the hypothesis and the conclusion.
Conditional Contrapositive Converse Inverse $p$ $q$ $~p$ $~q$ $p\to q$ $~q\to ~p$ $q\to p$ $~p\to ~q$ T T F F T T T T T F F T F F T T F T T F T T F F F F T T T T T T - The conditional statement is logically equivalent to the contrapositive.
- The converse is logically equivalent to the inverse.
- Know how to construct and use truth tables to determine whether statements are logically equivalent.

### 2.6 De Morgan’s Laws

- De Morgan’s Law for the negation of a disjunction states that, $~(p\vee q)$ is logically equivalent to $~p\wedge ~q.$
- De Morgan’s Law The negation of a conjunction states that, $~(p\wedge q)\equiv ~p\vee ~q.$
- Use De Morgan’s Laws to negate conjunctions and disjunctions.
- The negation of a conditional statement, if $p$ then $q$ is logically equivalent to the statement $p$ and not $q$. Use this property to write the negation of conditional statements.
- Use truth tables to evaluate De Morgan’s Laws.

### 2.7 Logical Arguments

- A logical argument uses a series of facts or premises to justify a conclusion or claim. It is valid if its conclusion follows from the premises, and it is sound if it is valid, and all of its premises are true.
- The law of detachment is a valid form of a conditional argument that asserts that if both the conditional, $p\to q$
Law of Detachment Premise: $p\to q$ Premise: $p$ Conclusion: $\therefore \mathrm{}q$ - Know how to apply the law of detachment to determine the conclusion of a pair of statements.
- The
Law of Denying the Consequent Premise: $p\to q$ Premise: $~q$ Conclusion: $\therefore \mathrm{}~p$ - Know how to apply the law of denying the consequent to determine the conclusion for pairs of statements.
- The chain rule for conditional arguments is a valid form of a conditional argument that asserts that if the premises of the argument have the form, $p\to q$
Chain Rule for Conditional Arguments Premise: $p\to q$ Premise: $q\to r$ Conclusion: $\therefore \mathrm{}p\to r$ - Know how to apply the chain rule to determine valid conclusions for pairs of true statements.