![Cartesian Product Cartesian Product](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHGK3RCkoRk0AGDePxNTYxfK5qNef28NOOs7hYvUvcNu86HYvup5cgftPWlUVlHMxBWoHM1jTPreRd4ZTXH9Ee1YMihoJAzAEzS5DxhJfWpUPA362YhJjVyJHBlVjy7KZod98Z7y8iwYIbo1wwqDTF0qFsVsQiQS9qoZIKSZ4zK989cG9n_SHJ9AWfbA/s16000/Cartesian%20Product.png)
If A and B are two non-empty sets then the set of all possible
ordered pairs (x, y) such that the first component x is an element of A and the
second component y is an element of B is called the Cartesian Product of set A
and B. It is denoted by A × B which is read as “A cross B”.
In the set-builder form, we can write
A × B = {(a, b): a ∈ A, b ∈ B}
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For example:
1.
If A = {a, b} and B = {1,
2} then, A × B = {(a, 1), (a, 2), (b, 1), (b, 2)} and B × A = {(1, a), (1, b),
(2, a), (2, b)}.
2.
If A = {1, 2, 3} and B =
{2, 4, 6} then, A × B = {(1, 2), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3,
2), (3, 4), (3, 6)} and B × A = {(2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4,
3), (6, 1), (6, 2), (6, 3)}.
3.
If A = {H, T} then A × A =
{(H, H), (H, T), (T, H), (T, T)}.
Note:
1.
A × B ≠ B × A.
2.
If m = number of elements
of A and n = number of elements of B, then the number of elements of A × B is
mn.
Representation of the Cartesian Product
If A = {1, 2, 3} and B = {1, 2} then the Cartesian Product A × B
can be represented as follows:
(a) Set of Ordered Pairs: A × B = {(1, 1), (1, 2), (2, 1,
(2, 2), (3, 1), (3, 2)}
(b) Lattice Diagram:
![Lattice Diagram Lattice Diagram](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifHMcyQ8sESVhpdaFsz47yDwgr8BXHvYCvXedN7A40Ddwftn-eb3pFqQZaRCDSmfyHkgnFsl53xUiBDGGUaFTn5yeVYbNo7OXFH3UYl9ZRZOsZqQl8hXgE5MLqib2tTMk9WzGGRXXMgZmFi7VZ9i3udz8S2gacWfCujP9Y2YoDTVJzbvQsxqJztpH7Rg/s16000/Lattice%20Diagram.png)
(c) Table:
![Table Table](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAmLQbOAVtbJmtQwguN2y9pWNgD7E77fYUqVpTDhg7TyXMsy84ZUyMHnmIKDSYBBPf5HnRquV_yL-_wriT_CnnYEwWSQtLjxHdKmovmviEXplX_6qz0i0VIqHEmKIUM3ELuCYUCJOShOPxfw5IXKxuFi1l8cNY4xpfdyVxa7-CQryZAns7zwQDTDx7yQ/s16000/Table.png)
(d) Mapping Diagram:
![Mapping Diagram Mapping Diagram](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTOZMfp2tYeEFAjB9Eo61IACvdq_2wSClafIUiZyOuJ40gCk0BKYg4xBc2RSYUPNsqYdKqrvsS1NhPfugMxT8RPepVvJn3V6dQz5nlLoDXzz-XxC-MBu2G_koHbp_AliZbEullq3Z2W_GR0a316yUdkU9yU6Q9GhnF-5bHglMketfcKfsgJ3LXYCz_Mw/s16000/Mapping%20Diagram.png)
(e) Tree Diagram:
![Tree Diagram Tree Diagram](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRtYGC5n439jgpCZiFEXIxnB3-asOy7gZzji-3gtrXnrU1dLngiUoBDFzzw93Rn_Gn7kGavwWj0v0YhbtwLQtINWvT7lF2tI9DZPlwi2IjcyylilnkANnDpGcXamaRXi4hwc6QyQahJgEPRIm28huWCRQpPvxfxTvl9Is8oDdjTZmTYnF1QMh4T4aOkg/s16000/Tree%20Diagram.png)
Worked Out Examples
Example 1: If A = {a, b} and B = {1, 2, 3} then find A × B.
Solution: Here, A = {a, b} and B = {1, 2, 3}
![Tree Diagram Example 1: Tree Diagram](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDfKfbZ_ZCZ_ELV-OSMxX7x1jIFVkl5IflZGHOOe483Qvm_3REXyZSk0-M5jynyJ9oXhAvnaBr-KO-pI5AUqS_9dKmz31xr9ZSPvnZfuBqmIexgDfZzl90VytR1vmdLBD3zwE8iCN42_YrPDprA-7XPx6cSiLWSYpCIk7rvxNFt30OMSMSmBjBoJn4_g/s16000/Example%201%20Tree%20diagram.png)
∴ A × B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}
Example 2: If A = {x ∈ N : x ≤ 4}, find A × A.
Solution: Here, A = {x ∈ N : x ≤ 4} ∴ A = {1, 2, 3, 4}
![Table Example 2: Table](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipVffAK00xZKsy0EgZw0udnkqIi1pFl3ZqiOsv3jgAUQiqULk0K5qnJgGb7Bf6UIR2QuYbcIYebK9wqvufeMH89AsJucq0y0Jo1JOnrlyKi26NCOZA5t3LtwBSmUCo1jyTFl6viiVkUva79gdI_CPxBREhrmKrA2M5xXvIirgj-ZfCa1TZjNbvs8XcAA/s16000/Example%202%20Table.png)
Now, from the above table
A × B = {(1, 1), (a, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3),
(2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4)}
Example 3: Draw the arrow diagram of A × B and B × A if A = {a, b} and B = {x,
y, z}.
Solution: Here, A = {a, b} and B = {x, y, z}
A × B in mapping diagram
![A × B in mapping diagram A × B in mapping diagram](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiv0Fe1ZNeed7EPdDycO_WkJ5BM0NtvLMO8uHYps0N1bjkf6bNWhH-YKG7iU-vUO-QgJITuRJqaRPcKDijFyuIuZasSgWJWM_vwrjo9oRPbsthVVWsj5IHsP6ym2AM6I1Px0dUChfbP5rX5Gl1FmDXOh1RleFyCvCmAP4hVgEloOPsE3wCeBlFOfpMg-w/s16000/A%20%C3%97%20B%20in%20mapping%20diagram.png)
A × B = {(a, x), (a, y), (a, z), (b, x), (b, y), (b, z)}
B × A in mapping diagram
![B × A in mapping diagram B × A in mapping diagram](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0II1CUEvbtW7hV_B2sWAT6VJTihw3HcsDFSt2HeSH24md7uoQ8AUKaqZsiXTlWwHY9AlPFgupCQWywfoIEYS-DgTy8gim-neu2G6tNnhJNjLQRXZMU71BxoAvRudokKnpApOvDJRy1TkqGmChEUwibbH61zt2B0ByqX5Tv054hDI6NJLganByGj2w4Q/s16000/B%20%C3%97%20A%20in%20mapping%20diagram.png)
B × A = {(x, a), (x, b), (y, a), (y, b), (z, a), (z, b)}
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