Transitions to Advanced Mathematics

Denial of a Proposition

A denial of a proposition, say proposition $P$, is any form of a proposition that is equivalent to $\neg P$ or $\neg P$. A proposition may only have one negation, but many denials.

Family

A family is a set of set, or sometimes known as a collection of sets. Families are often denoted by script letters, such as $\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}, \mathcal{E}, ...$. An example of a family could be something like: $\mathcal{A} = \{\{1, 2, 3\},\{\frac{4}{5}, \emptyset\},\{12, 61, 0.5\},\{1, 61, 32\}\}$.

Nonconstructive Proof

A nonconstructive proof is a proof of a conjecture that has the form "$(\exists x)P(x)$". The proof is specific in the sense that, to prove such a conjecture, we use a method that eludes to the existence of a element that satisfies $P(x)$ even though we do not know what that element is. For example, the usage of the Intermediate Value Theorem to know that there exists a solution where $x=0$ if there exists a upward concavity and downward concavity.

Proof of a Biconditional Statement

A proof of a biconditional statement, like $P\iff Q$ will have to employ a proof of $[P\implies Q]\wedge [Q\implies P]$. Hence, often times, a proof of a biconditional statement would result in two parts.

Direct Proofs

A proof of an implication ($P\implies Q$), that attempts to prove that if the antecedent ($P$) is true, then the consequent ($Q$) is also true. In the direct proof, we do not care about whether or not the antecedent is false since the implication is always true if the antecedent is false.

Theorem

A statement that describes a pattern among quantities or structures. These are statements asserted with existing proofs (in other words, theorems have proofs). Most of the time, theorems are non-self-evident statements.

Set-Builder Notation

An alternative method to denote the elements of a set. Set-builder notation is used for practicality and convenience when writing out sets containing a large amount of elements. The set containing all even numbers may be written as $\{x: x=2k \wedge k\in \mathbb{Z}\}$. We read the colon as "such as".

DeMorgan's Law of Set Theory

An extension of DeMorgan's laws for sets is as follows: 1. $(A\cup B)^c = A^c\cap B^c$. 2. $(A\cap B)^c = A^c\cup B^c$.

Negation of Quantified Statements

As proven in Theorem 1.3.1, the negation of a quantified statement is as follows: If $A(x)$ is an open sentence with variable $x$, then: 1. $\neg (\forall x)P(x) \iff (\exists x)\neg P(x)$ 2. $\neg (\exists x)P(x) \iff (\forall x)\neg P(x)$

Axioms

Axioms, also called postulates, are a set of statements assumed to be true. Mathematicians my deny an axiom and created other branches of mathematics. For example, in Euclidean Geometry, the statement "$a^2+b^2=c^2$, is true if $a$ and $b$ are side lengths of the segment of a right triangle not being the hypotenuse, and if $c$ is the longest side", is actually false depending on whether the axioms of Euclidean Geometry are accepted or not. Rejecting an axioms can often provide absurd mathematics, but also one which, perhaps, might be more real and useful.

Consistent Axiomatic System

Despite the impression that every conjecture or proposition may be proven in a well defined mathematical system with a set of axioms and definitions, there are some propositions that may not be proven. Such is the case with Euclid's fifth axiom of Euclidean Geometry. Consistent Axiomatic Systems referred to propositions that are proved impossible to be proven true or false.

Elements

Elements are objects of a set. EX: $A = \{1, 2, car, train\}$ has four distinct elements.

DeMorgan's Law of Indexed Family of Sets

Extending DeMorgan's laws, it is observed that for an indexed family of sets, say $\mathcal{A}$, we have: 1. $(\bigcap_{\alpha \in \Delta} A_{\alpha})^c = \bigcup_{\alpha \in \Delta} A_{\alpha}^c$. 2. $(\bigcup_{\alpha \in \Delta} A_{\alpha})^c = \bigcap_{\alpha \in \Delta} A_{\alpha}^c$.

Power Sets

For any set, say $A$, the power set of $A$ is the set whose elements are subsets of $A$ and it is denoted by $\mathcal{P} (A)$. Hence, symbolically, $\mathcal{P} (A) = {B:B\subseteq A}$. Note, the power set always contains the empty set, not ${\emptyset}$. This is because of the properties of sets. Also note, the power set contains $2^n$ elements, where $n$ is the number of elements in the original set. Sometimes the power set of, say set $A$ is denoted by $2^A$.

Properties of Sets

For every set, the following holds: 1. For every set, say $A$, $\emptyset \subset A$. 2. For every set, say $A$, $A\subseteq A$. 3. For all sets, $A$, $B$, $C$, if $A\subseteq B$ and $B\subseteq C$, then $A\subseteq C$.

Proper Subsets

For two sets, say $A$ and $B$, $A$ is a proper subset of $B$ if $A\subseteq B$ and $A\neq B$. We denote this by $A\subset B$. Otherwise, we say $A\subseteq B$.

Equivalence of Sets

For two sets, say $A$ and $B$, the sets are equal if and only if, all the elements in set $A$ is all the elements in set $B$. Hence, symbolically, we write $(\forall x)(x\in A\iff x\in B)$. Hence, for proving equivalence of sets, we have to prove $A\subseteq B$ and $B\subseteq A$

Indirect Proofs

Indirect proofs are proofs of another conjecture that is logically equivalent to the conjecture attempting to be proved, and hence proving the conjecture itself as well. Proof techniques that are indirect proofs include a proof by contraposition and a proof by contradiction.

Modus Ponens

Modus Ponens is an inference rule for conditional statements which states $P \wedge [P\implies Q\therefore Q]$, where the symbol $\therefore$ means "therefore". This says, if $P$ and $P\implies Q$ are both true, then $Q$ is automatically true.

Proofs

Proofs are the justifications for the truth of a theorem. Often times, proofs employ a systematic technique by imposing a logical form and solving the logical statement to show that the statement is always true/false.

Atmoic Propositions

Propositions that are not expanded by logical connectives. Simple propositions that can't be express with an additional logical connective.

Form of a Proposition

Recall, a proposition has exactly one truth value. The form of a proposition does not have a truth values. For example, $P\wedge Q$ can be a form of a proposition, but that does not have one truth value unless a proposition is assigned to $P$ and $Q$. Instead a form of a proposition has several truth values which depend on the truth values assigned to its subatomic propositions. We represent this by a truth table.

Hierarchy of Logical Connectives

Similar to how operations have a hierarchy, which we often denote as PEMDAS, there is also a hierarchy of logical connectives. TBC.

Relations

Simply put, a relation is an ordered pair. More rigorously, suppose that $A$ and $B$ are sets. Then, we say $R$ is a relation from $A$ to $B$ if and only if $R$ is a subset of $A\times B$.

Associative Laws of Set Theory

The associative laws of set theory state that: 1. $A\cup (B\cup C) = (A\cup B)\cup C$. 2. $A\cap (B\cap C) = (A\cap B)\cap C$.

Associative Laws

The associative laws states that for propositions $P$, $Q$, and $R$, the following are true: 1. $P\vee (Q\vee R)\equiv (P\vee Q)\vee R$ 2. $P\wedge (Q\wedge R)\equiv (P\wedge Q)\wedge R$

Commutative Laws of Set Theory

The commutative laws of set theory state that: 1. $A\cup B = B\cup A$ 2. $A\cap B = B\cap A$.

Commutative Laws

The commutative laws states that for propositions $P$ and $Q$, the following are true: 1. $P\wedge Q\equiv Q\wedge P$ 2. $P\vee Q\equiv Q\vee P$

The Complement of a Set

The complement of a set is simply everything besides the set. Suppose that $U$ is the universe. Then the complement is $U-A$ and is denoted by $A^c$.

Contrapositive

The contrapositive of an implication (or conditional), say $P\implies Q$ is the proposition $\neg Q\implies \neg P$. Consider that the original proposition $P\implies Q$ is logically equivalent to it's contrapositive. Hence, we can prove $P\implies Q$ by proving $\neg Q\implies \neg P$ (often known as prove by contraposition).

Converse

The converse of an implication (or conditional), say $P\implies Q$ is the proposition $Q\implies P$.

Double Negation Law

The double negation law states that $\neg \neg P \equiv P$.

Existential Quantifier

The existential quantifier is an attachment denoted by $\exists$. When placed in front of a open sentence, say $P(x)$ to get $(\exists x)P(x)$, it is read, "there exists $x$ such that $P(x)$". The sentence $(\exists x)P(x)$ is true if and only if the truth set of $P(x)$ is nonempty.

DeMorgan's Laws

The following laws, coined DeMorgan's laws in honor of DeMorgan, states that for propositions $P$ and $Q$ the following are true: 1. $\neg (P\wedge Q)\equiv \neg P\vee \neg Q$ 2. $\neg (P\vee Q)\equiv \neg P\wedge \neg Q$

Intersection of Sets

The intersection of sets, say set $A$ and set $B$, is the set of all elements that are in $A$ AND $B$. We denote the intersection of $A$ and $B$ by $A\cap B$. Hence, symbolically, we say $A\cap B=\{x: x\in A\text{ and } x\in B\}$.

Negation

The negation of a proposition is denoted by ~ or sometimes $\neg$ and applied in front of the proposition negating (e.g.) $\neg P$. The negation of the proposition $P$ is true exactly when $P$ is false. Often times we express the negation in english as "it is not the case...".

Universe of Discourse

The objects available for consideration as replacements for the variables in an open sentence. The universe is often expressed by number systems in mathematics, such as: $\mathbb{Z}, \mathbb{C}, \mathbb{N}, \mathbb{R}, \text{or } \mathbb{Q}$. Note that for the positive or negative values pertaining to a particular number system, say real numbers, it may be denoted as $\mathbb{R^+} \text{or } \mathbb{R^-}$

Equivalence of Open Sentences

Two open sentences, say $P(x)$ and $Q(x)$, are known to be equivalent predicates if and only if $P(x)$ has the same truth set as $Q(x)$ in EVERY universe possible.

Propositional Equivalence

Two propositions are equivalence if and only if they have the same truth values (truth tables).

Disjoint Sets

Two sets are known to be disjoint if and only if they have no elements in common. Hence, for sets $A$ and $B$, $A$ and $B$ are disjoint sets if $A\cap B=\emptyset$.

Undefined Terms

Undefined terms refer to an initial set of terms, in relation to a particular study, that mathematicians leave undefined to avoid circular definitions. They are often times referred to primitive notions in geometry (such as a point, which has no interpretation assigned to it). Circular definitions are convoluted interpretations that makes no progression. (E.g) An oak tree is a tree produced by an acorn. An acorn is what produces an oak tree.

Conjunction of $P$ and $Q$

Given the propositions $P$ and $Q$, the conjunction of $P$ and $Q$, denoted by $P\wedge Q$, is the proposition "$P$ and $Q$". $P\wedge Q$ is true exactly when both $P$ and $Q$ is true. Do not define the conjunction of $P$ and $Q$ as "a logical connective...", you are then defining only "$\wedge$". The english words, "but", "while", and "although", usually convey the use of a conjunction.

Subset

A subset is a set that where all elements in that set are contained in a given set. Hence, for two sets, $A$ and $B$, we say $A$ is a subset of $B$, and denote $A\subseteq B$, if and only if all elements in $A$ are also in $B$. With symbols, we write the definition as: $A\subseteq B\iff (\forall x)(x\in A\implies x\in B)$.

Truth Tables

A truth table for a proposition is a table consisting of atomic propositions, and simple compounded propositions, building up to the proposition desired. A truth table has all possible outcomes of every proposition building up to the desired proposition for every combination of truth values for the atomic propositions.

Inductive Set

A set, say $S$, of natural numbers whose property is that if $n\in S$, then $(n+1)\in S$ is called an inductive set.

Biconditionals

A biconditional proposition (or biconditional statement) is a proposition of the form "$P$ if and only if $Q$". A biconditional proposition is true only when $P$ and $Q$ has the same truth values. Sometimes the biconditional statement (or proposition) is expressed as $P$ iff $Q$. Note, two propositional forms are equivalent when $P \implies Q$ is a tautology. A biconditional statement with propositions $P$ and $Q$ is denoted as: $P\iff Q$. Alternative ways to express the biconditional: 1. $P$ if and only if $Q$. 2. $P$ if, but only if, $Q$. 3. $P$ is equivalent to $Q$. 4. $P$ is necessary and sufficient for $Q$.

Conditional

A conditional proposition is the proposition of the form "If $P$, then $Q$". Conditionals are often regarded as implications. In a conditional proposition, the atomic proposition $P$ is called the antecedent, and the proposition $Q$ is called the consequent. Conditional statements/propositions are denoted as $P\implies Q$. The above is only true if and only if $P$ is false or $Q$ is true. An elaboration on when it is true: If $P$ is false (like above) and $Q$ is true (like above), then the implication is true. If $P$ is false (like above) and $Q$ is false (unlike above), then the proposition is true. If $P$ is true (unlike above) and $Q$ is true (like above), then the implication is true. HOWEVER, if $P$ is true (unlike above) and $Q$ is false (unlike above), then the implication is false. Alternative ways to express the conditional: 1. $P$ implies $Q$. 2. $P$ is sufficient for $Q$. 3. $P$ only if $Q$. 4. $Q$ if $P$. 5. $Q$ whenever $P$. 6. $Q$ is necessary for $P$. 7. $Q$, when $P$.

Constructive Proof

A constructive proof is a proof of a conjecture that has the form "$(\exists x)P(x)$". The proof is specific in the sense that, to prove such a conjecture, we just name at least one element such that it satisfies the open sentence $P(x)$.

Proof by Exhaustion

A proof by examining every possible case (actually, proving every case). Proof by exhaustion is only reasonable when there is a finite amount of cases that is not physically or mentally straining to prove every single case. Often times this type of proof is employed in conjectures involving a potential bounded truth set containing integers.

Proof by Contraposition

A proof of contraposition proofs a conjecture by making use of the tautology $(P\implies Q)\iff (\neg Q\implies \neg P)$. Recall, that a biconditional also means equivalence. Hence, instead of proving $P\implies Q$, the statement $\neg Q\implies \neg P$ is proven instead, and thus we conclude that $P\implies Q$. Note, this is an indirect proof, but it is valid because of logical equivalences.

Proof by Contradiction

A proof that makes use of the tautology $P\iff [(\neg P)\implies (Q\wedge \neg Q)]$ for some proposition $Q$. Recall, that a biconditional also means equivalence. Hence, instead of proving $P\wedge Q$, the statement $\neg P\implies (Q\wedge \neg Q)$ is proven instead. Essentially, a proof by contradiction is saying that if $\neg P$ was true, then the assumption would be impossible, hence $Q\wedge \neg Q$. Thus, our assumption that $\neg P$ was true has to be false. An outline might be: Suppose $P$, therefore $Q$. Therefore, $\neg Q$. Hence $Q\wedge \neg Q$, which is a contradiction. An often overused, but quintessential, example is that $\sqrt{2}$ is irrational. More often than not, it is proved by first assuming that it IS rational.

Proposition Delimiters

A propositional delimiter is one of the following pairs of characters: )(, ][, or }{. Propositional delimiters are used to prevent ambiguities in compound propositions. Additionally we can add a "nor" for it is not the case for $P$ and it is not the case for $Q$. Or, $\neg (P\wedge Q)$, for "nor".

Contradiction

A propositional form that always yields a truth value of $F$ regardless of any assignments to the subatomic propositions.

Tautology

A propositional form that always yields a truth value of $T$ regardless of any assignments to the subatomic propositions.

Empty Set

A set containing no elements. The general empty set is true for every domain or universe. The empty set is denoted by $\emptyset$.

Set

A set is a collection of objects that we call elements, or members of the set. Consider, if we have an object $x$ in a set called $A$, then we are permitted to write $x\in A$. In all instances otherwise, we say $x\notin A$. Sets are denoted by brackets followed by whatever elements are in the set.

Union of a Family

A union of a family of sets is the set containing all the elements of every set in a particular family of sets. Thus, for the family $\mathcal{A}$, we symbolically denote the union of the family of $\mathcal{A}$ as $\bigcup_{A\in \mathcal{A}} A$, where $\bigcup_{A\in \mathcal{A}} A = \{x: x\in A \text{ for SOME } A\in \mathcal{A}\}$. From the definition, we may state that $x\in \bigcup_{A\in \mathcal{A}} A$ if and only if $(\exists A\in \mathcal{A})(x\in A)$.

Without Loss of Generality

An assumption in a proof that makes it possible to prove a theorem by reducing the number of cases to consider in the proof. Often times it means the ensuing cases are proved in the same manner. An example: Prove that if there are two integers $m$ and $n$, one that is even and one that is odd, then $m^2+n^2$ has the form $4k+1$ for some integer $k$. When proving, we may say: "Let $m$ and $n$ be integers. Without loss of generality, we may assume that $m$ is even and $n$ is odd...

Intersection of a Family

An intersection of a family of sets is the set containing all the elements that every set in a particular family of sets contain. Thus, for the family $\mathcal{A}$, we symbolically denote the intersection of the family of $\mathcal{A}$ as $\bigcap_{A\in \mathcal{A}} A$, where $\bigcap_{A\in \mathcal{A}} A = \{x: x\in A \text{ for ALL } A\in \mathcal{A}\}$. From the definition, we may state that $x\in \bigcap_{A\in \mathcal{A}} A$ if and only if $(\forall A\in \mathcal{A})(x\in A)$.

Open Sentences

An open sentence is a statement that contains variables, hence creating an ambiguous truth value. An alternative name for "open sentences" are predicates. Predicates become propositions with there is an assignment to the variables or a quantifier attached to the open sentence. An example of an open sentence is: $x\leq 3$, since we don't know the value of $x$. We can assign similar notation as function notation to predicates. When we have a open sentence with the variable $x$, the predicate (or open sentence) may be denoted as $P(x)$.

Unary Operations

An operation that only involves one operand. For example, $1 + 1 = 2$ is a unary operation. However, expressions with multiple operations (like $1+3\times 4^2$) are not unary operations.

Ordered Pair of The Form $(a, b)$.

An ordered pair of the form $(a, b)$ is an object formed by two elements $a$ and $b$.

Important Theorems of Set Theory

Below is an important list of theorems used in set theory. For sets $A$, $B$, and $C$, the following holds: 1. $A\subseteq A\cup B$. 2. $A\cap B\subseteq A$. 3. $A\cap \emptyset = \emptyset$. 4. $A\cup \emptyset = A$. 5. $A\cap A = A$. 6. $A\cup A = A$. 7. $A-\emptyset = A$. 8. $\emptyset - A = \emptyset$. 9. $A\subseteq B\iff A\cup B=B$. 10. $A\subseteq B\iff A\cap B=A$. 11. If $A\subseteq B$, then $A\cup C\subseteq B\cup C$. 12. If $A\subseteq B$, then $A\cap C\subseteq B\cap C$. 13. Commutative Laws of Set Theory. 14. Associative Laws of Set Theory. 15. Distributive Laws of Set Theory. 16. $A-B=A\cap B^c$. 17. $(A^c)^c=A$. 18. $A\cup A^c = U$. 19. $A\cap A^c = \emptyset$. 20. $A\subseteq B\iff B^c\subseteq A^c$. 21. $A\cap B = \emptyset \iff A\subseteq B^c$.

Inverse of a Conditional

Given a conditional statement or proposition (for example $P\implies Q$, the inverse of a conditional is defined to be $\neg P\implies \neg Q$. In other words, the inverse of a conditional is the negation of both the antecedent and consequent.

Disjunction of $P$ and $Q$

Given the propositions $P$ and $Q$, the disjunction of $P$ and $Q$, denoted by $P\vee Q$ is the proposition "$P$ or $Q$". $P\vee Q$ is true when exactly one of $P$ or $Q$ is true. Do not define the disjunction of $P$ and $Q$ as "a logical connective...", you are then defining only "$\vee$".

Theorems for Family of Sets

Here are some important theorems pertaining to the study of a family of sets: For any arbitrary family of sets, say $\mathcal{A}$: 1. $\bigcap_{A\in \mathcal{A}} A\subseteq B$. 2. $B\subseteq \bigcup_{A\in \mathcal{A}} A$. 3. If the family $\mathcal{A}$ is nonempty, then $\bigcap_{A\in \mathcal{A}} A\subseteq \bigcup_{A\in \mathcal{A}} A$

Propositions

Sentences with exactly one truth value, either the sentence is true or false, which we denote by T or F.

Conjecture

Similar to theorems, proofs describe a pattern among quantities or structures, however conjectures do not have a proof. Conjectures are often unproven statements that mathematicians would like to prove.

Properties of Cross Product

Some of the properties of taking the cross product of two sets are observed below, for the sets $A$, $B$, $C$, and $D$: 1. $A\times (B\cup C) = (A\times B)\cup (A\times C)$. 2. $A\times (B\cap C) = (A\times B)\cap (A\times C)$. 3. $A\times \emptyset = \emptyset$. 4. $(A\times B)\cap (C\times D) = (A\cap C)\times (B\cap D)$. 5. $(A\times B)\cup (C\times D)\subseteq (A\cup C)\times (B\cup D)$. 6. $(A\times B)\cap (B\times A) = (A\cup B)\times (A\cup B)$.

Principle of Mathematical Induction

Suppose that $S$ is a subset of $\mathbb{N}$ (the natural numbers), with the following properties: 1. $1\in S$ 2. For all $n\in \mathbb{N}$, if $n\in S$, then $(n+1)\in S$. Then, $S = \mathbb{N}$ by the Principal of Mathematical Induction. Sometimes the principle of mathematical induction (abbreviated as PMI) is used to defined sets. However, for the most part, we show that if $S$ is a set with certain properties, then $S$ is equal to the set of natural numbers, meaning that the set of natural numbers have that property, hence meaning that the property is true for all natural numbers.

Indexed Family of Sets

Suppose that $\Delta$ is a nonempty set and that for each element of $\Delta$, say $\alpha$, where $\alpha \in \Delta$, there exists a corresponding set, denoted by $A_{\alpha}$. Then, the set $\{A_{\alpha}: \alpha \in \Delta\}$ is called an indexed family of sets. Additionally, for such a case, the set $\Delta$ is called an indexing set, where $\alpha \in \Delta$ is called an index.

Cross Product of Sets

Suppose that we have two sets, say $A$ and $B$, the cross product, denoted by $A\times B$ is the set of all ordered pairs that can be formed from an element in set $A$ and an element in set $B$. Hence, symbolically, we say $A\times B = \{(a, b): [a\in A]\wedge [b\in B]\}$. The new set is sometimes referred to as the "Cartesian Product" in honor of Descartes.

Difference of Sets

The difference of sets, say set $A$ and set $B$, is the set of all elements that are unique to $A$. In other words, it is the set of all elements that are in $A$ but not in $B$ or $A\cap B$. We denote the difference of set $A$ and $B$ as $A-B$. Symbolically, we say, $A-B=\{x:x\in A\wedge x\notin B\}$.

Distributive Laws of Set Theory

The distributive laws of set theory state that: 1. $A\cup (B\cap C) = (A\cup B)\cap (A\cup C)$. 2. $A\cap (B\cup C) = (A\cap B)\cup (A\cap C)$

Distributative Laws

The distributive laws states that for propositions $P$, $Q$, and $R$, the following are true: 1. $P\wedge (Q\vee R)\equiv (P\wedge Q)\vee (P\wedge R)$ 2. $P\vee (Q\wedge R)\equiv (P\vee Q)\wedge (P\vee R)$

Law of Excluded Middle

The law of excluded middle says that the sentence "any statement is either true or false" is always true, and is always a tautology. This means $P\vee \neg P$ is always true.

Properties of Natural Numbers

The set of natural numbers has some very distinct properties, namely: 1.

Truth Set

The truth set is a collection of objects that may be assigned to a variable in a predicate to make the predicate become a proposition with a truth value of T.

Union of Sets

The union of two sets, say set $A$ and set $B$, is the set of all elements that are in $A$ OR $B$. We denote the union of $A$ and $B$ by $A\cup B$. Hence, symbolically, we say $A\cup B=\{x: x\in A \text{ or } x\in B\}$.

Unique-Existential Quantifier

The uniqueness quantifier (or unique-existential quantifier) is an attachment denoted by $\exists !$. When placed in front of an open sentence, say $P(x)$ to get $(\exists !x)P(x)$, it is read, "there exists a unique $x$ such that $P(x)$". The sentence $(\exists !x)P(x)$ is true if and only if the truth set contains exactly one element. The unique-existential quantifier is

Universal Quantifier

The universal quantifier is an attachment denoted by $\forall$. When placed in front of an open sentence, say $P(x)$ to get $(\forall x)P(x)$, it is read, "for all $x$, $P(x)$". The sentence $(\forall x)P(x)$ is true if and only if the entire universe is the truth set. In casual English conversations, this might be expressed as "for each", "every", etc. Sometimes it is implied, such as "Polynomial functions are continuous", which is actually intended to be more correctly stated as "All polynomial functions are continuous".