Cartesian Product & Ordered Pairs of Sets-Properties, FAQs & Solved Problems | Class 11 Math Notes Study Material Download Free PDF

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Cartesian Product is also one such operation that is performed on two sets, which returns a set of ordered pairs.

In this article, we have covered, the ordered pair definition, a cartesian product of sets, and others in detail.

What is an Ordered Pair?

An ordereยญd pair has two parts. The first part is called the first componeยญnt. The second part is called theยญ second component. We writeยญ an ordered pair like this: (a, b). Theยญ letter โ€˜aโ€™ is the first componeยญnt. The letter โ€˜bโ€™ is theยญ second component. An ordereยญd pair has two things. One thing comes first. The otheยญr thing comes second.

Example:

(5, 7) is an ordered pair of integers.

Note: (5, 7) โ‰  (7, 5), an ordered pair (a, b) is equal to (x, y) only if a = x and b = y.

Cartesian Product of Sets

When two seยญts have items in them, A and B, theยญir Cartesian product is all the pairs you can make. Oneยญ part of the pair comes from set A. Theยญ other part comes from set B. Weยญ make every possibleยญ pair this way. The result is a new seยญt of all these pairs. We writeยญ this new set as Aร—B.

A ร— B = {(a, b) : a โˆˆ A and b โˆˆ B}

Example: 

Let A = {1, 2} and B = {4, 5, 6}

A ร— B = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6)}

Here the first component of every ordered pair is from set A the second component is from set B.

Cartesian Product of two sets can be easily represented in the form of a matrix where both sets are on either axis, as shown in the image below. Cartesian Product of  A = {1,  2} and B = {x, y, z}

Cartesian product & ordered pairs of sets-properties, faqs & solved problems | class 11 math notes study material download free pdf

Properties of Cartesian Product

Various properties of cartesian product includes,

1. Cartesian Product is non-commutative: A ร— B โ‰  B ร— A

Example: 

A = {1, 2} , B = {a, b}

A ร— B = {(1, a), (1, b), (2, a), (2, b)}

B ร— A = {(a, 1), (b, 1), (b, 1), (b, 2)}

Therefore as A โ‰  B we have A ร— B โ‰  B ร— A

2. A ร— B = B ร— A, only if A = B

Proof:

Let A ร— B = B ร— A then we have  

A โІ  B  and B โІ  A, it follows that A = B

3. Cardinality of Cartesian Product is defined as number of elements in A ร— B and is equal to the product of cardinality of both sets i.e.,

|A ร— B| = |A| ร— |B|

Proof:

Let a โˆˆ A then the number of ordered pair (a, b) such that b โˆˆ B is |B|

Therefore we have |B| choices for b for each a where a โˆˆ A therefore the number of element in A ร— B is |A| ร— |B|

4. A ร— B = โˆ…, if either A = โˆ… or B = โˆ…

Proof: 

Suppose Aร—B=โˆ…. This means there are no ordered pairs (a,b) where aโˆˆA and bโˆˆB.

If A is non-empty, then there exists at least one element aโˆˆA. For any such a, there should be an ordered pair (a,b) for some bโˆˆB, as B is not empty. But since we have assumed Aร—B=โˆ…, this is a contradiction. Hence, A must be empty.

Similarly, if B is non-empty, then there exists at least one element bโˆˆB. For any such b, there should be an ordered pair (a,b) for some aโˆˆA, as A is not empty. But since we have assumed Aร—B=โˆ…, this is a contradiction. Hence, B must be empty.

Therefore, if ? ร— ? = โˆ…, either A or B must be empty

Hence, the statement ? ร— ? = โˆ… if and only if either A=โˆ… or ? = โˆ… is proven.

Problems on Cartesian Product of Sets

Problem 1: Find the value of x and y given (2x โ€“ y,  25) = (15,  2x + y)?

Solution:  

As we know from the property of ordered pairs, 2x โ€“ y = 15 and 25 = 2x + y.

Solving the linear equations we have x = 10 and y = 5.

Problem 2. Given A = {2, 3, 4 , 5} and B = {4 , 16 , 23}, a โˆˆ A, b โˆˆ B, find the set of ordered pairs such that a2 < b?

Solution:

As 22 < 16 and 23, 32 < 16 and 23, 42 < 23  

We have the set of ordered pairs such that a2 < b is {(2, 16), (2, 23), (3, 16), (2, 23), (4, 23)}

Problem 3. If A = {9, 10} and B = {3, 4, 6}, find A ร— B and |A ร— B|? 

Solution:

A ร— B = {(9, 3), (9, 4), (9, 6), (10, 3), (10, 4), (10, 6)}

|A ร— B| = |A| * |B| = 2 * 3 = 6

Problem 4. If A ร— B = {(a, x), (a, y ), (b, x ), (b, y)}, find A and B?

Solution:

We know A is the set of all first components in ordered pairs of A ร— B and 

B is the set of the second component in the ordered pair of A ร— B.

Therefore A = {a, b} and B = {x, y}

Problem 5. Given A ร— B has 15 ordered pairs and A has 5 elements, find the number of elements in B?

Solution:

We know |A ร— B| = |A| * |B|, 15 = 5 * |B|

Therefore B has 15 / 5 = 3 elements.

Conclusion

The Cartesian Product of Sets is a fundamental concept in set theory and mathematics that helps in understanding the combination of elements from the two or more sets. By creating ordered pairs from the elements of the sets it provides a structured way to explore relationships and combinations. The practice problems presented above illustrate the various scenarios where the Cartesian Product can be applied ranging from the simple sets to more complex combinations.

FAQs on Cartesian Product of Sets

What is cartesian product of two sets i.e. A ร— B?

Cartesian product of sets A and B, denoted Aร—B, is the set of all possible ordered pairs where the first element is from A and the second from B.

Define ordered pair.

An ordered pair is a pair of elements (a, b) in which the order of the elements is significant. This means (a, b) is distinct from (b, a) if a is not equal to b.

What is cartesian product of 3 sets?

Cartesian product of three sets A, B, and C is the set of all possible ordered triples where first element is from A, second from B, and third from C.

Write formula for cartesian product of sets.

Cartesian product of two sets A and B is defined as:

A ร— B = {(a, b) | a โˆˆ A and b โˆˆ B}

For three sets A, B, and C:

A ร— B ร— C = {(a, b, c) | a โˆˆ A, b โˆˆ B, and c โˆˆ C}

What is cartesian product of a set and a null set?

Cartesian product of a set A and a null set (โˆ…) is always an empty set (โˆ…), as there are no elements in the null set to form pairs with elements from set A.

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