3.8 Cartesian product

We have seen that set operations convey the notion of arithmetic operations. One such similar operation is product of two sets called “Cartesian product”. Since sets are collection – not a single quantity, the product operation here involves combining or pairing each of the elements of one set with that of another set.

We use symbol “X” to denote product operation. The Cartesian product of two sets “A” and “B” is symbolically represented as :

$$A\times B$$

It is important to understand that we do not multiply elements as we do in arithmetic – instead we pair elements together. This is the meaning of “product” for the sets. We denote one such pair within a pair of small brackets like :

$$\left(a,b\right)$$

where $a\in A$ and $b\in B$ .

Note that elements from two sets are separated by comma.

Ordered pair

The order of pairing is important. The pair (a,b) and (b,a) are different. This ordering is required as there are real time situations, where order makes a difference. Consider for example, we are required to find the integers which can be formed from two integer subsets like {1,2,3} and {3,4,5}. Clearly, “13” and “31” represent different integers. We need to distinguish them. All pairs formed from two sets should be distinct.

Keeping this restriction in mind, let us work out an example to find ordered pairs formed from elements of two sets.

$$A=\text{set of first letter of the names of cities}=\{N,D,H\}$$

$$B=\text{set of numbers denoting flight numbers}=\left\{\mathrm{001,002,003}\right\}$$

All possible ordered pairs formed from two sets are :

$$\left(N,001\right),\left(N,002\right),\left(N,003\right),\left(D,001\right),\left(D,002\right),\left(D,003\right),\left(H,001\right),\left(H,002\right),\left(H,003\right)$$

There are all together 9 ordered pairs. From this example, we can deduce a method for writing ordered pairs from two sets. We begin with the first elements of two sets. Progressively, we change the elements from the second set till it is exhausted, while keeping the elements from the first set unchanged. Then, we switch to “next” element from first set and start with “first” from the second set. Again, we change the elements from the second set progressively till it is exhausted, while keeping the elements from the first set unchanged. We continue in this manner till all elements from the first set is also exhausted.

From this discussion, it is also evident that two ordered pairs are equal if and only if the corresponding first and second elements are equal.

Cartesian product

The Cartesian product of two sets is defined in terms of ordered pairs.

Cartesian product

The Cartesian product of two non-empty sets “A” and “B” is the set of all ordered pairs of the elements from two sets.

We should emphasize the use of word “non-empty”, The Cartesian product of a non-empty set with an empty set is equal to empty set.

$$A\times \phi =\phi$$

On the other hand, if one of the sets is infinite, then resulting Cartesian product is also infinite.

We express the Cartesian product set in set building form as :

$$A\times B=\{\left(x,y\right):x\in A,y\in B\}$$

Here, use of "," in the set builder form is equivalent to "and". Therefore, we can write Cartesian product of two sets also as :

$$A\times B=\{\left(x,y\right):x\in A\mathrm{and}y\in B\}$$

Further, we can emphasize two ways validity of the conditional statements as in the case of other set operators :