Cartesian Product Calculator

About Cartesian Product Calculator

The Cartesian product of two sets is the set of all ordered pairs where the first element is from A and the second from B.

How to use this calculator

1. Enter the first set, for example \(A = \{1,2\}\).
2. Enter the second set, for example \(B = \{x,y\}\).
3. Calculate to list every ordered pair \((a,b)\) with \(a \in A\) and \(b \in B\).
4. Check the result to make sure each element from the first set is matched with every element from the second set.

The formula explained

The formula \(A \times B = \{(a,b) \mid a \in A, b \in B\}\) means the Cartesian product is the set of all ordered pairs where the first entry comes from \(A\) and the second entry comes from \(B\). The order matters, so \((1,x)\) is different from \((x,1)\).

• A = the first set
• B = the second set
• a = an element chosen from set \(A\)
• b = an element chosen from set \(B\)
• (a,b) = an ordered pair made from one element of \(A\) and one element of \(B\)

Step by step method

1. List the elements of the first set and the second set.
2. Pair each element of the first set with every element of the second set.
3. Write the pairs in ordered form, with the first coordinate from the first set and the second coordinate from the second set.

Worked example

Problem. Find the Cartesian product of \(A = \{1,2\}\) and \(B = \{3,4\}\).

1. Start with the first element of \(A\), which is 1, and pair it with every element of \(B\): \((1,3)\), \((1,4)\).
2. Now use the second element of \(A\), which is 2, and pair it with every element of \(B\): \((2,3)\), \((2,4)\).
3. Combine all the ordered pairs into one set: \(A \times B = \{(1,3),(1,4),(2,3),(2,4)\}\).

Tips and common mistakes

• Do not switch the order of the sets unless you want a different product, because \(A \times B\) is not the same as \(B \times A\).
• Remember that the output is a set of ordered pairs, so repeated pairs should appear only once.

Frequently asked questions