Sets – Cartesian Product of Sets

If A and B are two non empty sets, then their Cartesian Product A x B is set of all possible ordered pairs.

A x B = {(x, y) : x ∈A, y ∈ B}

where the elements of A are comes first and the elements of B are comes second.

from all Sets A and B, written as A x B, is expressed as

(A x B) = {(x, y)}

where x is element in A, y is element in B.

Example 1: Suppose A = {tea, coffee}, and B = {milk, water}

The Cartesian Product of Sets A and B (A x B) = {(tea, milk), (tea, water), (coffee, milk), (coffee, water)}

Example 2: If A = {1, 2} and B = {a, b, c}. Find cartesian product of A x B

The Cartesian Product of Sets A and B (A x B) is {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}

Example 3: If A = {x, y, z} and B = {1, 2, 3}. Find cartesian product of A x B

Here, A = {x, y, z} and B = {1, 2, 3}

A x B = {(x, 1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3), (z, 1), (z, 2), (z, 3)}

If X be the set of points on x-plane and Y be the set of points on y-plane then, X x Y represents the on XY plane.

Example 4: If A = {1, 2} and B = {4, 5, 6}. Find cartesian product of A x B

The Cartesian Product of Sets A and B (A x B) is {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6)},

Example 5: If A = {a, b} and B = {1, 2, 3}. Find cartesian product of A x B and B x A

Here, A = {a, b} and B = {1, 2, 3}

A x B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}

B x A = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}

Example 6: If A x B = {(x, 1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3)}. Find A and B

Set A contains first element of each ordered pair only,
A = {x, y}

Set B contains second element of each ordered pair only,
B = {1, 2, 3}

A set can contain only unique element, so we exclude duplicate elements in both sets.

A = {x, y} and B = {1, 2, 3}

Example 7: If A x B = {(p, x), (p, y), (q, x) (q, y). Find A and B.

Set A contains first element of each ordered pair only,
Thus A = {p, q}

Set B contains second element of each ordered pair only,
B = {x, y}

A set can contain only unique element, so we exclude duplicate elements in both sets.

A = {p, q,} and B = {1, 2, 3}

Example 8: If A = {x, y} and B = {1, 2, 3, 4, 5}. Find cartesian product of A x B and B x A

Here, A = {x, y} and B = {1, 2, 3, 4, 5}

A x B = {(x, 1), (x, 2), (x, 3), (x, 4), (x, 5), (y, 1), (y, 2), (y, 3), (y, 4), (y, 5)}

B x A = {(1, x), (2, x), (3, x), (4, x), (5, x), (1, y), (2, y), (3, y), (4, y), (5, y))}

Here, we note that cartesian product of A x B ≠ B x A,

i.e. the cartesian product is not commutative.

Example 9: If A and B are two sets, and A x B consists 6 elements. If three elements of A x B are (1, 4), (2, 3) and (5, 3). Find A x B.

Here, elements of A x B are (1, 4), (2, 3) and (5, 3)

so, {1, 2, 5} are elements of A and {4, 3} are elements of B.

Now A x B = {(1, 4), (1, 3), (2, 4), (2, 3), (5, 4), (5, 3)}.

Thus, A x B contain six ordered pairs.

Condition for commutative property

Two sets A and B, the Cartesian product of A x B and B x A are equal

(i) If we have two finite sets A and B, where A has X elements and B has Y elements, then A x B has (X x Y) elements.

This rule is very useful in counting elements in a set. This rule is called the multiplication principal rule.

The number of elements in a set is denoted by A, so here we write A = X, B = Y and AB = XY.

In above example, A = 3, B = 2, thus A x B = 3 x 2 = 6.

(ii) If either of the following conditions are satisfied, then Two sets A and B, the Cartesian product of A x B and B x A are equal.

If A = (a, b) and B = ∅,

Then, A x B = ∅ and B x A = ∅

Hence, A x B = B x A

If A = (1, 2) and B = (1, 2), then

A x B = {(1, 1), (1, 2), (2, 1), (2, 2)}

B x A = {(1, 1), (1, 2), (2, 1), (2, 2)}

A x A = A² = {(1, 1), (1, 2), (2, 1), (2, 2)}

B x B = B² = {(1, 1), (1, 2), (2, 1), (2, 2)}

Hence, A x B = B x A = A² = B²

Non Commutativity property

If two sets A and B are unique and non-empty, then A x B is not equal to B x A.

If A = (p, q) and B = (1, 2, 3) then,

A x B = {(p, 1), (p, 2), (p, 3), (q, 1), (q, 3), (q, 3)}

B x A = {(1, p), (1, q), (2, p), (2, q), (3, p), (3, q)}

Here, A x B ≠ B x A

Cartesian Product of Empty set

If two either of two set A or B are null sets, then cartesian product of A x B will also be an empty set,

i.e., if A = ∅ or B = ∅, then A x B = ∅

Example 10: A x ∅ = ∅, since no ordered pairs can be formed when one of the sets is empty.

If A = (a, b) and B = .∅

Then A x B = ∅ and B x A = ∅

If either of two set is empty, then the Cartesian Product of these two sets is also an empty set.

Cartesian product of three sets

If we have three finite sets A, B and C, then the cartesian product of A, B and C is denoted by A x B x C and define as:

(A x B X C) = {(a, b, c) a ∈A, b ∈ B and c ∈ C}

Example 11: Set A = (1, 2), B = (x, y) and C = (3, 4). Find A x B x C

Here, A = (1, 2), B = (x, y) and C = (3, 4)

Now, A x B = {(1, x,), (1, y), (2, x ), (2, y)}

A x B x C = {(1, x,), (1, y), (2, x ), (2, y)} x {3, 4}

(A x B x C) = {(1, x, 3), (1, x, 4), (1, y, 3) (1, y, 4), (2, x, 3 ), (2, x, 4), (2, y, 3), (2, y, 4)}

Example 12: If A = {1, 2, 5} and B = {1, 2}. Find cartesian product of (1) A x B (2) B x A (3) A x A (4) B x B

Here, A = {1, 2, 5} and B = {1, 2}

(A x B) = {1, 2, 5} x {1, 2} = {(1 , 1), (1 , 2), (2, 1), (2, 2), (5, 1), (5, 2)}

(B x A) = {1, 2} x {1, 2, 5} = {(1, 1), (1, 2), (1, 5), (2, 1), (2, 2), (2, 5)}

(A x A) = {1, 2, 5} x {1, 2, 5} = {(1 , 1), (1 , 2), (1, 5), (2, 1), (2, 2), (2, 5), (5, 1), (5, 2), (5, 5)}

(B x B) = {1, 2} x {1, 2} = {(1 , 1), (1 , 2), (2, 1), (2, 2)}

Example 13: If A = {1, 2, 3} and B = {5, 2}. Find cartesian product of (1) A x B (2) B x A (3) A x A (4) B x B

Here, A = {1, 2, 3} and B = {5, 2}

(A x B) = {1, 2, 3} x {5, 2} = {(1 , 5), (1 , 2), (2, 5), (2, 2), (3, 5), (3, 2)}

(B x A) = {5, 2} x {1, 2, 3} = {(5, 1), (5, 2), (5, 3), (2, 1), (2, 2), (2, 3)}

(A x A) = {1, 2, 3} x {1, 2, 3} = {(1 , 1), (1 , 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

(B x B) = {5, 2} x {5, 2} = {(5 , 5), (5 , 2), (2, 5), (2, 2)}

Example 14: If A = {a, b, c} and B = {b, c}. Find cartesian product of (1) A x B (2) B x A (3) A x A (4) B x B

Here, A = {a, b, c} and B = {b, c}

(A x B) = {a, b, c} x {b, c} = {(a , b), (a , c), (b, b), (b, c), (c, b), (c, c)}

(B x A) = {b, c} x {a, b, c} = {(b, a), (b, b), (b, c), (c, a), (c, b), (c, c)}

(A x A) = {a, b, c} x {a, b, c} = {(a , a), (a , b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)}

(B x B) = {b, c} x {b, c} = {(b , b), (b , c), (c, b), (c, c)}

(A x B) (B x A)= {(a, b, c) x (b, c) (b, c) x (a, b, c)} = {(a , b), (b , a), (a, c), (c, a), (b, b), (b, c), (c, b), (c, c)}

(A x B) (B x A)= {(a, b, c) x (b, c) (b, c) x (a, b, c)} = {(b , b), (b , c), (c, b), (c, c)}

Example 15: If A = {x, y}, B = {a, b} and C = {b, c}. Find cartesian product of (1) A x B (2) B x A (3) A x A (4) B x B

Here, A = {x, y} and B = {a, b} and C = {b, c}

(A x B) = {x, y} x {a, b} = {(x , a), (x , b), (y, a), (y, b)}

(A x C) = {x, y} x {b, c} = {(x, b), (x, c), (y, b), (y, c)}

B ⋃ C = (a, b) ⋃ (b, c) = (a, b, c)

A x (B⋃C) = {x, y} x {a, b, c} = {(x , a), (x , b), (x, c), (y, a), (y, b), (y, c)}

(A x B) = {x, y} x {a, b} = {(x , a), (x , b), (y, a), (y, b)}

(A x C) = {x, y} x {b, c} = {(x, b), (x, c), (y, b), (y, c)}

(A x B) ⋃ (A x C) = {(x, a}, {x, b), (x , a), (x , c), (y, a), (y, b), (y, c)}

(A x C) = {x, y} x {b, c} = {(x, b), (x, c), (y, b), (y, c)}

Example 16: If A = {x, y}, B = {a, b} and C = {b, c}. Find value of (1) B ∩ C (2) A x (B ∩ C) (3) (A x B) ∩ (A x C)

Here, A = {x, y}, B = {a, b} and C = {b, c}

B ∩ C = (a, b) ∩ (b, c)

B ∩ C = (b)

A x (B ∩ C) = (x, y) x (b)

A x (B ∩ C) = {(x, b), (y, b)}

(A x B) = (x, y) x (a, b)

(A x B) = {(x, a), (x, b), (y, a), (y, b)}

(A x C) = (x, y) x (b, c)

(A x C) = {(x, b), (x, c), (y, b), (y, c)}

(A x B) ∩ (A x C) = {(x, b), (y, b)}