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A well-defined group of items or elements is called a set. The Cartesian product of sets is the combination of all the possible ordered pairs of sets. There are different important theorems on the Cartesian product of sets used to solve the complex set theory problems.
The important theorems on the Cartesian product of sets explain the different properties of the Cartesian product. The Cartesian product of two non-empty sets, X and Y, is expressed below.
XY={(x,y) |x X and yY}
The theorem which explains the significance of the order of sets in the Cartesian product is the Non-Commutative theorem.
The Cartesian product
The two-dimensional coordinate plane, which is built by the two different intersection axes, is called the Cartesian. One of the two axes of Cartesian is the x-axis, and the other one is the y-axis. The Cartesian product of sets implies the product of two elements of Cartesian in a systematic manner. The Cartesian product of two sets is a cross product or the product of two sets, say X and Y.
The Cartesian product of two non-empty sets, X and set Y, is expressed by the following expression.
XY
Consider set X{x1,x2}and set Y{y1,y2}. The Cartesian product of these two sets is expressed by the following expression.
XY={(x1y1) ,(x1y2) ,(x2,y1), (x2y2)}
Important theorems on the Cartesian product of sets
There are different important theorems on the Cartesian product of sets. These theorems are very helpful in understanding the set theory of the Cartesian product of sets. The important theorems of the Cartesian product of sets are tabulated below.
Theorem | Property | Expression |
Theorem 1 | Non-Commutative | AB BA |
Theorem 2 | Non-Associative | (AB )CA(BC) |
Theorem 3 | Distributive | A(BC )=(AB)(AC) A(BC )=(AB)(AC) A(B-C )=(AB)-(AC) |
Theorem 4 | Null Set | A= |
Theorem 5 | Subset | ACBC |
Theorem 1
AB BA
The Cartesian product of sets is non-commutative. This theorem suggests that the result of the Cartesian product is dependent on the order of the sets. Let two non-empty sets be A and B. Then the Cartesian product of set A and set B is not equal to the Cartesian product of set B and set A.
AB BA
Proof: Suppose set A{a} consists of an element a and set B consists and element b. In this case, the Cartesian product of set A and set B is, {a,b} and the Cartesian product of set B and set A is {b,a}.
{a,b} {b,a}
AB BA
The Cartesian product of set A and set B is not equal to the Cartesian product of set B and set A except A=B.
For the two equal sets, A=B. This theorem can be changed in the following way.
AA =A2
Theorem 2
(AB )CA(BC)
The Cartesian product of sets is non-associative. Take three non-empty sets, A, B and C. For the Cartesian product of set A and set B, the non-associative property can be given by the following expression.
(AB )CA(BC)
Proof: Suppose set A {a} consists of an element a, set B consists of an element b and set C consists of an element c. In this case, the Cartesian product of set A and set B is {a,b} and the Cartesian product of set B and set A is {b,a}.
{(a,b),c} {a(b,c)}
(AB )CA(BC)
Theorem 3
Distributive theorem of the Cartesian product:
A(BC )=(AB)(AC)
A(BC )=(AB)(AC)
A(B-C )=(AB)-(AC)
Distributive theorem of intersection: The Cartesian product of set A and the intersection of set B and set C is equal to the intersection of the Cartesian product of set A and B and the Cartesian product of set A and C.
A(BC )=(AB)(AC)
Distributive theorem of union: The Cartesian product of set A and union of set B and set C is equal to the union of the Cartesian product of set A and B and union of the Cartesian product of set A and C.
A(BC )=(AB)(AC)
Distributive theorem of difference: Distributive property of difference of set A, set B and set C for the Cartesian product is expressed below.
A(B-C )=(AB)-(AC)
Theorem 4
A=
A set that does not contain any element is called a null set. The symbol which is used to represent the null set is called the phi, and it looks like . In the Cartesian product, if any set is a null set, then the result of the Cartesian product is also a null set. Let’s suppose there is a non-empty set A and a null set B. The Cartesian product of these sets can be expressed by the following expression.
A=
For sets A and B, this theorem can also be represented in the following way.
AB=
If either A= or B=
Theorem 5
ACBC
The Cartesian product of two sets A and Set C is a subset of the Cartesian product of two sets, set B and set C, if set A is a subset of B.
If, AB then,
ACBC
Conclusion
The important theorems on the Cartesian product of sets are used in set theory while performing the Cartesian product of sets. The non-commutative theorem of the Cartesian product of sets is the most significant theorem, which suggests that the result of the Cartesian product is dependent on the order of the sets.
