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Operations on Sets: Union, Intersection, Difference, Complement, Symmetric Difference, Cartesian product

operations on set in maths- union, intersection, difference, complement

Basic Operations on Sets –

There are certain operations that, when performed on two sets, result in the creation of another set. Now we will define some set operations and examine their properties to see how they work. Union of Sets, Intersection of Sets, Difference of Sets, Symmetric Difference of Sets, Complement of Set, and Cross Product or Cartesian product of Set, are included in Set operations. All of our sets will now be referred to as subsets of some universal set.

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1. Union of Sets:

Let A and B be any two sets. The union of A and B denoted by $A \cup B$ , is the set of all elements which belongs to A or belongs to B or both sets, that is,

$A \cup B$ = $\{ x:x \in A$ or $x \in B\}$

For example, Let A = {1, 2, 3, 4} and B = {3, 1, 6, 8}, then,

$A \cup B$ = {1, 2, 3, 4, 6, 8}.

Note: In $A \cup B$ , we write all the elements of A and all the elements of B, but the common elements of Sets A and B have been taken only once.

2. Intersection of Sets:

Let A and B be any two sets. The intersection of A and B denoted by $A \cap B$ , is the set of all common elements of A and B. In other words, the intersection of A and B is the set of all elements which belongs to both A and B, that is,

$\[A \cap B$ = $\{ x:x \in A$ and $x \in B\}$

For example, Let A = {1, 2, 3, 4} and B = {3, 1, 6, 8}, then,

$A \cap B$ = {1, 3}.

3. Difference of Sets:

Let A and B be any two sets. The difference of A and B is the set of elements which belong to set A but not to set B. It is denoted by $A - B$ and reads ‘A minus B’. Then,

$A - B$ = $\{ x:x \in A$ and $x \notin B\}$

Similarly,

$B - A$ = $\{ x:x \in B$ and $x \notin A\}$

Therefore,

$A - B \ne B - A$

For example, Let A = {1, 2, 3, 4} and B = {3, 1, 6, 8}, then,

$A - B$ = {2, 4}, and $B - A$ = {6, 8}.

4. Symmetric Difference of Sets:

Let A and B be any two sets. The symmetric difference of A and B is defined as,

$A \oplus B = (A - B) \cup (B - A)$

For example, Let A = {1, 2, 3, 4} and B = {3, 1, 6, 8}, then,

$A - B$ = {2, 4}, and $B - A$ = {6, 8},

and so,

$A \oplus B = (A - B) \cup (B - A)$

$= (2,4) \cup (6,8)$

$= (2,4,6,8)$ .

5. Complement of Set:

Let U be the universal set and A is a subset of U. Then the complement of set A is the set of all elements of U that are not in A. It is denoted by ${A'}$ or ${A^c}$ . That is,

${A'}$ = ${A^c}$ = $\{ x:x \in U$ and $x \notin A\}$ .

Obviously,

${A'}$ = ${A^c}$ = $U - A$ .

For example, Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 2, 3, 4}, then,

${A'}$ = ${A^c}$ = $U - A$

= {0, 5, 6, 7, 8, 9}.

6. Cross Product or Cartesian product of Set:

Let A and B are two non-empty sets. The Cross product or Cartesian product of A and B, denoted by $A \times B$ is defined as,

$A \times B = \{ (a,b):a \in A$ and $b \in B\}$ .

Similarly,

$B \times A = \{ (b,a):a \in A$ and $b \in B\}$ .

The cross product $A \times B$ of the sets is the set of all possible order pairs where the first component belongs to set A and the second component belongs to set B.

For example, Let A = {1, 2} and B = { $a,b,c$ }, then,

$A \times B = { (1,a),\left( {1,b} \right),\left( {1,c} \right),(2,a),\left( {2,b} \right),\left( {2,c} \right)}$ , and

$B \times A = { (a,1),\left( {a,2} \right),\left( {b,1} \right),(b,2),\left( {c,1} \right),\left( {c,2} \right)}$ .

If either A or B are empty sets, then their Cartesian product is also the empty set, that is, if $A = \phi$ or $B = \phi$ , then

$A \times B = \phi$ .

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About Lata Agarwal 268 Articles

M.Phil in Mathematics, skilled in MS Office, MathType, Ti-83, Internet, etc., and Teaching with strong education professional. Passionate teacher and loves math. Worked as a Assistant Professor for BBA, BCA, BSC(CS & IT), BE, etc. Also, experienced SME (Mathematics) with a demonstrated history of working in the internet industry. Provide the well explained detailed solutions in step-by-step format for different branches of US mathematics textbooks.

Operations on Sets: Union, Intersection, Difference, Complement, Symmetric Difference, Cartesian product

Basic Operations on Sets –

1. Union of Sets:

2. Intersection of Sets:

3. Difference of Sets:

4. Symmetric Difference of Sets:

5. Complement of Set:

6. Cross Product or Cartesian product of Set:

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Basic Operations on Sets –

1. Union of Sets:

2. Intersection of Sets:

3. Difference of Sets:

4. Symmetric Difference of Sets:

5. Complement of Set:

6. Cross Product or Cartesian product of Set:

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