In the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If $x\in A$ and $A\not\subset B$, then $x\in B$

A set is a well-defined and distinct collection of objects and it is denoted by capital letters like $A,B,C...$

2. Representation of Sets:

There are two methods of representing a set,

(i) Roster or Tabular form:

We list all the members of the set within braces $\left\{\right\}$ and separate by commas.

For example: $A=\left\{1, 2, 3, 4\right\}$

(ii) Set-builder form:

We list the property or properties satisfied by all the elements of the sets.

For example: $A=\left\{x: x\in N, x<5\right\}$

3. Types of Sets:

(i) Empty Set:

A set consisting of no element is called the Empty set and is denoted by $\varphi .$

(ii) Singleton Set:

A set consisting of a single element is called a singleton set.

(iii) Finite and infinite Set:

A set consisting of a finite number of elements is called a finite set, otherwise the set is called an infinite set.

(iv) Equal Sets:

Two sets $A$ and $B$ are equal if they have exactly the same elements.

(v) Equivalent Sets:

Two finite sets $A$ and $B$ are said to be equivalent if the number of elements are equal, i.e. $n\left(A\right)=n\left(B\right)$

(vi) Universal Set:

The set of all elements of all related sets. It is usually denoted by the symbol $U$.

(vii) Disjoint Sets:

Two sets which do not have any element in common are called Disjoint Sets. 

4. Subset & Superset:

(i) Subset:

A set $A$ is said to be a subset of a set $B$, if every element of $A$ is also an element of $B$ i.e, $A\subseteq B$, if $x\in A\Rightarrow x\in B$.

(ii) Superset:

A set $B$ is said to be the Super set of a set $A$, if every element of $A$ is also an element of $B$.

5. Some Important Points:

(i) Every set is a subset of itself.

(ii) The empty set is a subset of every set.

(iii) The total number of subsets of a finite set containing $n$ elements is $2^n$

6. Intervals as Subsets of $\mathit{R}$:

If $a,b$ are real numbers such that $a<b,$ then the set

(i) Closed Interval:

$\left\{x:x\in R\right.$ and $\left.a\le x\le b\right\}$ is called the closed interval $\left[a, b\right]$

(ii) Open Interval:

$\left\{x:x\in R\right.$ and $\left.a<x<b\right\}$ is called the open interval $\left(a, b\right)$

(iii) Semi-open or Semi-closed Interval:

(a) $\left\{x:x\in R\right.$ and $\left.a\le x<b\right\}$ is called the semi-open or semi-closed interval $\left[a,\hspace{0.17em}b\right).$

(b) $\left\{x:x\in R\right.$ and $\left.a<x\le b\right\}$ is called the semi-open or semi-closed interval $\left(a,\hspace{0.17em}b\right].$

7. Power Set:

The collection of all subsets of a set $A$ is called the power set of $A$ and is denoted by $P\left(A\right)$.

8. Operations of Sets:

(i) Union of two sets:

The union of two sets $A$ and $B$ is the set of all those elements which are either in $A$ or in $B$ or in both and is denoted by $A\cup B.$ Thus, $A\cup B=\left\{x:x\in A\hspace{0.17em}\mathrm{or}\hspace{0.17em}x\in B\right\}.$

(ii) Intersection of two sets:

The intersection of two sets $A$ and $B$ is the set of all those elements which are common to both $A$ and $B$ and is denoted by $A\cap B.$ 

(iii) Difference of sets:

The difference $A-B$ of two sets $A$ and $B$ is the set of all those elements of $A$ which do not belong to $B$ i.e. $A-B=\left\{x:x\in A\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}x\notin B\right\}.$.Similarly, $B-A=\left\{x:x\in B\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}x\notin A\right\}.$

(iv) Symmetric difference of two sets:

The symmetric difference of two sets $A$ and $B$ is the set $\left(A-B\right)\cup \left(B-A\right)$ and is denoted by $A\Delta B.$

(v) Complement of a set:

The complement of a subset $A$ of universal set $U$ is the set of all those elements of $U$ which are not the elements of $A$. The complement of $A$ is $A\text{'}$ or $A^c$.

9. Laws of Algebra of Sets:

For any three sets $A,B$ and $C$, we have

(i) $A\cup A=A$ and $A\cap A=A$ (Idempotent laws)

(ii) $A\cup \varphi =A$ and $A\cap U=A$ (Identity laws)

(iii) $A\cup B=B\cup A$ and $A\cap B=B\cap A$ (Commutative laws)

(iv) $\left(A\cup B\right)\cup C=A\cup \left(B\cup C\right)$ and $\left(A\cap B\right)\cap C=A\cap \left(B\cap C\right)$ (Associative laws)

(v) $A\cup \left(B\cap C\right)=\left(A\cup B\right)\cap \left(A\cup C\right)$ and $A\cap \left(B\cup C\right)=\left(A\cap B\right)\cup \left(A\cap C\right)$ (Distributive laws)

(vi) $\left(A\cup B\right)\text{'}=A\text{'}\cap B\text{'}$ and $\left(A\cap B\right)\text{'}=A\text{'}\cup B\text{'}$ (De Morgan's laws)

10. Formulae to Solve Practical Problems on Union and Intersection of Two Sets:

If $A,B$ and $C$ are finite sets and $U$ be the finite universal set, then

(i) $n\left(A\cup B\right)=n\left(A\right)+n\left(B\right)-n\left(A\cap B\right)$

(ii) $n\left(A\cup B\right)=n\left(A\right)+n\left(B\right)\iff A, B$ are disjoint non-void sets

(iii) $n\left(A-B\right)=n\left(A\right)-n\left(A\cap B\right)$ i.e., $n\left(A-B\right)+n\left(A\cap B\right)=n\left(A\right)$

(iv) $n\left(A\Delta B\right)=n\left(B-A\right)+n\left(A-B\right)=n\left(A\right)+n\left(B\right)-2n\left(A\cap B\right)$

(v) $n\left(A\cup B\cup C\right)=n\left(A\right)+n\left(B\right)+n\left(C\right)-n\left(A\cap B\right)-n\left(B\cap C\right)-n\left(C\cap A\right)+n\left(A\cap B\cap C\right)$

(vi) Number of elements in exactly two of sets $A,B$ and $C$$=n\left(A\cap B\right)+n\left(B\cap C\right)+n\left(C\cap A\right)-3n\left(A\cap B\cap C\right)$

(vii) Number of elements in exactly one of sets $A,B$ and $C$$=n\left(A\right)+n\left(B\right)+n\left(C\right)-2n\left(A\cap B\right)-2n\left(B\cap C\right)-2n\left(A\cap C\right)+3n\left(A\cap B\cap C\right).$