Flashcards: Set Theory Fundamentals — 34 cards

What is a set according to Georg Cantor?

A collection of clearly distinguished objects considered as a whole.

What does the notation $a \in A$ signify?

That a belongs to set A.

What does the notation $a \notin A$ signify?

That a does not belong to set A.

How can a set be specified in roster form?

By listing its elements.

How can a set be specified in descriptive form?

By stating the condition satisfied by its elements.

When are two sets considered equal?

When they contain exactly the same elements.

Does the order of elements affect set equality?

No, order does not affect set equality.

What distinguishes a finite set from an infinite set?

A finite set has a listing that eventually stops, an infinite set never stops.

What is the finite set of natural divisors of 6?

The set $T_6=\{1,2,3,6\}$.

What is the infinite set of natural multiples of 6?

The set $V_6=\{6,12,18,\dots\}$.

What does the notation $A\subset B$ mean?

Every element of A also belongs to B.

What is B called if $A\subset B$?

B is the superset of A.

What is the empty set?

A set that contains no elements.

Is the empty set a subset of every set?

Yes, the empty set is a subset of every set.

What does the universal set G contain?

All objects allowed in the set discussion.

What is the relation between sets under consideration and the universal set G?

They are subsets of G.

How many subsets does a set with n elements have?

It has $2^n$ subsets.

What does the intersection $A\cap B$ represent?

Elements belonging to both A and B.

What does the union $A\cup B$ represent?

Elements belonging to A, B, or both.

What does the difference $A\setminus B$ represent?

Elements belonging to A but not to B.

What does the complement of A include?

All elements of the universal set G not in A.

What does the identity $A\cap A=A$ express in set algebra?

Intersecting a set with itself yields the same set.

What is the result of the set operation $A\cup A$?

The union of a set with itself is the same set.

What does the identity $A\setminus A=\varnothing$ mean?

Removing a set from itself results in the empty set.

What does the identity $A\cap(A\cup B)=A$ represent in set algebra?

It expresses absorption of union by intersection.

What does the identity $A\cup(A\setminus B)=A$ express?

It expresses absorption of set difference by union.

When are two sets called disjoint?

When their intersection is the empty set.

What formula gives the cardinality of the union of two finite sets?

$|A\cup B|=|A|+|B|-|A\cap B|$

What formula expresses the cardinality of the difference of two sets?

$|A\setminus B|=|A|-|A\cap B|$

How do you count elements satisfying at least one of two conditions?

Count each set, subtract their overlap, to get the union.

How do you convert a set from descriptive to roster form?

List every object satisfying the condition.

How do you convert a set from roster to descriptive form?

Identify the common property and express it as a condition.

How can the set $\{1,3,7,21\}$ be described?

As the set of natural divisors of 21.

How can the set $\{1,4,9,16,25,\dots\}$ be described?

As the set of natural square numbers.