CS173 UIUC Mathematics Terminology

Questions and Answers

What does the notation Z represent?

All complex numbers

All integers (correct)

All real numbers

All rational numbers

What does N denote in mathematics?

Non-negative integers including zero.

What does Z+ represent?

Positive integers not including 0.

What does R represent in mathematics?

Real numbers. Signup and view all the answers

What is the definition of Q in mathematics?

Rational numbers. Signup and view all the answers

What is C in the context of numbers?

Complex numbers. Signup and view all the answers

What does ε signify in strings?

The string containing no characters. Signup and view all the answers

What is concatenation of strings?

Combining two strings end-to-end. Signup and view all the answers

What is a proposition?

A statement that is either true or false. Signup and view all the answers

What does the expression p ∧ q represent?

The logical conjunction of p and q. Signup and view all the answers

What does p ∨ q represent?

The logical disjunction of p and q. Signup and view all the answers

What is the definition of exclusive or in logic?

True if either p or q is true, but not both. Signup and view all the answers

What does p → q mean?

If p, then q. Signup and view all the answers

What is the converse of a statement p → q?

q → p. Signup and view all the answers

What does p ↔ q indicate?

p if and only if q. Signup and view all the answers

What is the contrapositive of p → q?

¬q → ¬p. Signup and view all the answers

What does ¬(p ∧ q) simplify to?

¬p ∨ ¬q. Signup and view all the answers

What does ¬(p ∨ q) simplify to?

¬p ∧ ¬q. Signup and view all the answers

What does ¬(p → q) represent?

p ∧ ¬q. Signup and view all the answers

What is a universal quantifier?

For all... (∀x). Signup and view all the answers

What is an existential quantifier?

There exists... (∃y). Signup and view all the answers

What does ¬(∀x, P(x)) mean?

∃x, ¬P(x). Signup and view all the answers

What are general proof methods?

Universal and existential arguments. Signup and view all the answers

What does a | b indicate?

a divides b. Signup and view all the answers

What is gcd?

Greatest common divisor. Signup and view all the answers

What is the least common multiple (lcm)?

ab/gcd(a,b). Signup and view all the answers

If a=bq+r, what can we say about gcd?

gcd(a,b)=gcd(b,r). Signup and view all the answers

What does a ≡ b mod k mean?

k divides (a-b). Signup and view all the answers

What does reflexive mean in relation?

Every element is related to itself. Signup and view all the answers

What does irreflexive mean?

x is not related to itself for all x in A. Signup and view all the answers

What is a symmetric relation?

If xRy then yRx. Signup and view all the answers

What does antisymmetry mean?

If xRy and yRx, then x=y. Signup and view all the answers

What is a transitive relation?

If aRb and bRc, then aRc. Signup and view all the answers

What is a partial order?

Reflexive, antisymmetric, and transitive relation. Signup and view all the answers

What is a linear order?

A partial order where every pair of elements is comparable. Signup and view all the answers

What is a strict partial order?

Irreflexive, antisymmetric, and transitive relation. Signup and view all the answers

What defines an equivalence relation?

Reflexive, symmetric, and transitive. Signup and view all the answers

What is a basic function?

f:A→B where A is the domain and B is the co-domain. Signup and view all the answers

What does onto mean in terms of functions?

∀y ∈ B, ∃x ∈ A, f(x) = y. Signup and view all the answers

Study Notes

Sets and Numbers

Z: Represents all integers; includes {..., -3, -2, -1, 0, 1, 2, 3}.
N: Denotes non-negative integers; considered natural numbers including 0.
Z+: Indicates positive integers, excluding 0.
R: Denotes real numbers; includes all rational and irrational numbers.
Q: Represents rational numbers; designed as fractions p/q, with q ≠ 0, and includes operations for addition and multiplication.
C: Refers to complex numbers; represented in the form a + bi.

Strings and Propositions

ε in strings: Symbolizes the empty string, having a length of 0.
Concatenation of strings: Combines two strings, e.g., "blue" + "cat" equals "bluecat".
Proposition: A declarative statement that is either true or false; it cannot be a question or include variables.

Logical Operators

p ∧ q (And): Truth table shows that both statements must be true for the conjunction to be true.
p ∨ q (Or): Truth table indicates the disjunction is true if at least one statement is true.
Exclusive or: True if only one of p or q is true; false if both are true.
p → q (Implies): Truth table specifies that the implication is false only when p is true and q is false.
Converse: The converse of p → q is q → p; they are not equivalent.
Biconditional (p ↔ q): States p if and only if q; both must be true or both false.
Contrapositive: ¬q → ¬p; logically equivalent to the original statement.
De Morgan's Laws:
- ¬(p ∧ q) = ¬p ∨ ¬q
- ¬(p ∨ q) = ¬p ∧ ¬q
Negation of implications: ¬(p → q) rewrites to p ∧ ¬q.

Quantifiers

Universal Quantifier: Symbolized as ∀x; means "for all".
Existential Quantifier: Symbolized as ∃y; means "there exists".
Negation of universal statements: ¬(∀x, P(x)) is equivalent to ∃x, ¬P(x).

Proof Techniques

Universal proof methods: General arguments can be proved or disproved with specific counter-examples.
Existential proof methods: Specific examples can affirm existence, while general arguments can refute it.

Number Theory

Divisibility: a | b signifies that a divides b; b can be expressed as an = b for some integer n.
gcd (Greatest Common Divisor): Found by inspecting prime factors and identifying shared ones.
lcm (Least Common Multiple): Calculated as lcm(a,b) = ab/gcd(a,b).
GCD property: For any integers a and b, if a = bq + r, then gcd(a, b) = gcd(b, r).

Relations and Order

a ≡ b mod k: Indicates that k divides (a - b).
Reflexive Relation: Every element is related to itself, satisfied when for all x ∈ A, xRx holds.
Irreflexive Relation: No element is related to itself.
Symmetric Relation: Mutual relation where if xRy, then yRx.
Antisymmetric Relation: Distinct elements are never related in both directions; if xRy and yRx, then x must equal y.
Transitive Relation: If aRb and bRc, then a must relate to c.
Partial Order: A relation that is reflexive, antisymmetric, and transitive.
Linear Order: A partial order where every pair of elements is comparable.
Strict Partial Order: Characterized by being irreflexive, antisymmetric, and transitive.
Equivalence Relation: Defined as being reflexive, symmetric, and transitive.

Functions

Basic Function: Defined as f: A → B, where A is the domain and B is the co-domain. The number of possible functions is P^N, where P is elements in B.
Onto Function: For every y in B, there exists an x in A such that f(x) = y.

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Test your knowledge of mathematical sets with this quiz focused on fundamental number classifications. Designed for students in the CS173 course at UIUC, it covers integers, natural numbers, rational numbers, and real numbers. Challenge yourself and enhance your understanding of these essential concepts!