02 Sets and Set Operations Free Online FlashCards

What is a set?

A set is a well-defined collection of distinct objects called elements or members; order and repetition do not matter.

What does $A\subseteq B$ mean?

The notation $A\subseteq B$ means every element of $A$ is also an element of $B$.

How can set equality be proved?

Two sets are equal exactly when they contain the same elements. Equivalently, $A=B$ iff $A\subseteq B$ and $B\subseteq A$.

What does set union contain?

The union $A\cup B$ contains every element that belongs to $A$, to $B$, or to both.

What does set intersection contain?

The intersection $A\cap B$ contains exactly the elements common to both $A$ and $B$.

What is $A\setminus B$?

The difference $A\setminus B$ contains elements that are in $A$ but not in $B$. In general, $A\setminus B\ne B\setminus A$.

What is the complement $A^c$?

Relative to a specified universal set $U$, the complement is $A^c=U\setminus A$: all elements of $U$ that are not in $A$.

What is the Cartesian product $A\times B$?

The Cartesian product $A\times B$ is the set of ordered pairs $(a,b)$ with $a\in A$ and $b\in B$.

How large is $A\times B$?

For finite sets, $|A\times B|=|A||B|$. If $|A|=m$ and $|B|=n$, there are $mn$ ordered pairs.

What is the power set $\mathcal{P}(A)$?

The power set $\mathcal{P}(A)$ is the set of all subsets of $A$, including $\varnothing$ and $A$ itself.

How many subsets does an $n$-element set have?

If $|A|=n$, then $|\mathcal{P}(A)|=2^n$, because each element is either included in or omitted from a subset.

What does an indexed union represent?

An indexed union contains elements belonging to at least one family member: $\bigcup_{i\in I}A_i=\{x\mid\exists i\in I,\,x\in A_i\}$.