Set theory is the branch of mathematics that learned about the sets, which are the collections of objects. Even though any type of objects can be collected into a set, set theory is applied most often to objects that are related to mathematics.

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There are six basics operations in set theory. The names are,

**Union**

In basic set theory, the union of sets A and B is represented by A?B. That is group the values of the set A and set B.

Union of sets is the set defined as

A $\cup$ B = {a | a $\in$ A or a $\in$ B }

**Intersection**

The intersection of the sets A and B represented by A ? B. It contains common elements of the set A and set B.

Intersection of sets is the set defined as

A $\cap$ B = {a | a $\in$ A and a $\in$ B }

**Complement**

Complement of set A is represented by A^{c}, is contain the all elements of universal set that are not members of A.

**Difference**

The difference of sets is denoted by A - B. If we subtract set B from set A, then A - B is the set of all element in A, but not in B.

Difference of sets is the set defined as

A - B = { a | a $\in$ A and a $\notin$ B }

**Cartesian Product**

In basics set theory, the set A and set B represented by a A x B.

Difference of sets is represented by

A x B = {(a, b) | a $\in$ A and b $\in$ B}

**Powers Set**

Power set is the set whose elements are all possible subsets of given set.

**Idempotent Laws**

- A?A = A
- A?A = A

**Commutative Laws**

- A?B = B?A
- A?B = B?A

**Associative Laws**

(A?B) ? C = A?(B?C)

(A?B) ?C = A?(B?C)

**Distributive Laws**

- A?(B?C) = (A?B) ?(A?C)
- A?(B?C) = (A?B) ? (A?C)

**Identity Laws**

- A?Ø = A
- A?U = U
- A?Ø = Ø
- A?U = A

**Complement Laws**

- A?A’ = U
- (A’)’ = A
- A?A’ = Ø
- A - B = A?B’

**DeMorgan’s Laws**

- (A?B)’ = A’ ? B’
- (A?B)’ = A’ ? B’

**Consistency Principle**

- A`sube` B iff A?B = B.
- A`sube` B iff A?B = A.

**Prove that **

i) A?(B?C) = (A?B) ?(A?C)

ii) A?(B?C) = (A?B) ? (A?C)

**If A = {1, 3, 4, 5, 6}, B = {2, 3, 4, 5, 6} and C = {1, 6, 7, 8, 9}**

**Solution**

The given sets are A = {1, 3, 4, 5, 6}, B = {2, 3, 4, 5, 6} and C = {1, 6, 7, 8, 9}

**i) A?(B?C) = (A?B) ?(A?C)**

**First solve left hand side****,** A?(B?C)

**A?(B?C)** = { 1, 3, 4, 5, 6 }

The right side condition solve by the following way.

**(A?B) ?(A?C)**

A?B = {1, 2, 3, 4, 5, 6}

A?C = {1, 3, 4, 5, 6, 7, 8, 9}

(A?B) ? (A?C) = {1, 3, 4, 5, 6}

So A?(B?C) = (A?B) ? (A?C).

**ii) A ? (B?C) = (A?B) ? (A?C)**

Take left hand side condition

**A ? (B?C)**

B?C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

A ? (B?C) = {1, 3, 4, 5, 6}

Now take the right hand side condition .

**(A?B) ? (A?C)**

A?B = {3, 4, 5, 6}

A?C = {1, 6}

(A?B) ? (A?C) = {1, 3, 4, 5, 6}

So A ? (B?C) = (A?B) ? (A?C).

Hence proved.

More topics in Set Theory | |

Set Operations | Laws of Set Theory |

Cardinal Numbers | How Many Elements Are in the Power Set? |