Fundamentals of Set Theory and Logic

Null set

A set with no members.

Subset

Every member of A is in B.

Proper subset

A is a subset of B, but not vice versa.

Universal set

Set containing all discussed subsets.

Equal sets

Two sets with exactly the same members.

Equivalent sets

Two sets with the same number of elements.

Power set

Set of all subsets of A.

Finite set

Set with a fixed number of elements.

Infinite set

Set without a fixed number of elements.

List method

Elements written in braces, separated by commas.

Set-builder notation

Uses variables to define set elements.

Element notation

Indicates membership in a set (e.g., 1 ∈ A).

Cardinal number

Number of distinct elements in a set.

Rooster method

Listing elements explicitly in a set.

Description method

Describing properties of set elements.

Set notation

Symbols used to represent sets and their relationships.

Natural numbers

Positive integers starting from 1.

Odd integers

Integers not divisible by 2.

Even integers

Integers divisible by 2.

Set A

Example set containing {1, 2, 3}.

Set B

Example set of even natural numbers less than 10.

Set D

Set of numbers x such that x² = 25.

Membership symbol

Indicates if an element belongs to a set.

Singleton Set

A set with only one element.

Union of Sets

Elements in A, B, or both.

Intersection of Sets

Elements common to both A and B.

Difference of Sets

Elements in A but not in B.

Complement of Set

Elements in universal set not in A.

Disjoint Sets

Sets with no elements in common.

De Morgan's Laws

Relations between unions and intersections.

Idempotent Laws

A∪A = A and A∩A = A.

Associative Laws

Grouping does not affect union/intersection.

Commutative Laws

Order does not affect union/intersection.

Distributive Laws

Distributing intersection over union and vice versa.

Identity Laws

A∪∅ = A and A∩∅ = A.

Universal Set

Set containing all possible elements.

Venn Diagram

Visual representation of set relationships.

Set Operators

Symbols used to denote set operations.

Complement Laws

A∪A' = U and (A')' = A.

Total Elements (TE)

Total number of elements in combined sets.

Example of Sets

A={1,2,3}, B={3,4} shows union/intersection.

Set Notation

A={x | x belongs to A}.

Set Membership

Indicates if an element belongs to a set.

Set Representation

Using braces to denote elements in a set.

Real Numbers (R)

All numbers on the number line, including rationals.

Rational Numbers (Q)

Numbers expressible as a fraction of integers.

Irrational Numbers (I)

Numbers that cannot be expressed as fractions.

Integers (J)

Whole numbers including negatives, zero, and positives.

Whole Numbers (W)

Non-negative integers including zero.

Natural Numbers (N)

Positive integers starting from 1.

Domain of a Relation

Set of all first elements in ordered pairs.

Range of a Relation

Set of all second elements in ordered pairs.

Ordered Pair

A pair of elements in a specific order.

Cartesian Product

Set of all ordered pairs from two sets.

Function

A relation where each input has one output.

One-to-One Function

Distinct inputs map to distinct outputs.

Onto Function

Every element in range corresponds to an input.

Everywhere Defined Function

Domain includes all possible inputs.

Combination of Functions

Operations combining two or more functions.

Sum of Functions

(f + g)(x) = f(x) + g(x).

Difference of Functions

(f - g)(x) = f(x) - g(x).

Product of Functions

(f ∙ g)(x) = f(x) ∙ g(x).

Quotient of Functions

(f / g)(x) = f(x) / g(x).

Composition of Functions

(f g)(x) = f[g(x)].

Disjunction

Compound proposition true if at least one part is true.

Proposition

A declarative statement that is true or false.

Truth Value

Determines if a proposition is true or false.

Truth value

Indicates if a proposition is true or false.

Propositions

Statements that can be true or false.

Propositional variables

Lower-case letters representing arbitrary propositions.

Truth values

Designated as T (true) or F (false).

Negation

Reverses the truth value of a proposition.

Open sentences

Propositions using variables, not fixed truth values.

Conjunction

True if both propositions are true (p ∧ q).

Implication

True unless p is true and q is false.

Converse

Swaps antecedent and consequent in a conditional.

Inverse

Negates both antecedent and consequent of a conditional.

Contrapositive

Negates and swaps antecedent and consequent.

Disjunction

True if at least one proposition is true.

Biconditional

True when both propositions share the same truth value.

Compound proposition

Combines two or more propositions using connectives.

Connectives

Words that link propositions, like 'and' or 'or'.

Example of negation

Five is not an even number.

Example of conjunction

There are 12 months and Christmas is in December.

Example of implication

If birds can chirp, then birds cannot fly.

Example of equivalence

Eight is a cube iff 3 times 2 is 6.

False proposition

A statement that does not hold true.

True proposition

A statement that holds true.

Variables in logic

Symbols representing values in propositions.

Logical connectives

Operators that combine propositions in logic.

Truth Value

Indicates whether a proposition is true or false.

Propositional Function

A function that returns truth values based on propositions.

Tautology

True regardless of component propositions' truth values.

Contradiction

False for all possible truth value combinations.

Radix

Base of a number system, e.g., binary is base 2.

Binary Number

Number expressed using only 1s and 0s.

Bit

Smallest unit of information, either 0 or 1.

Most Significant Bit (MSB)

Leftmost bit with highest binary weight.

Least Significant Bit (LSB)

Rightmost bit with lowest binary weight.

Decimal System

Base 10 system using digits 0-9.