If $P imes Q =\{(4,5),(4,6),(4,7),(5,5),(5,6),(5,7)\}$, find P and Q . Also show that $n ( P imes Q )= n ( P ) imes n ( Q )$.

Question

If $P 	imes Q =\{(4,5),(4,6),(4,7),(5,5),(5,6),(5,7)\}$, find P and Q . Also show that $n ( P 	imes Q )= n ( P ) 	imes n ( Q )$.

GinnyAnswer · Answer

Find set P by collecting the first elements of the ordered pairs: P = { 4 , 5 } . 
 Find set Q by collecting the second elements of the ordered pairs: Q = { 5 , 6 , 7 } . 
 Calculate the number of elements in $P ", 
 
 Explanation 
 
 Analyze the problem We are given the Cartesian product $P ", 
 
 Finding set P The set P consists of all the first elements in the ordered pairs of $P ", 
 
 Finding set Q The set Q consists of all the second elements in the ordered pairs of $P ", 
 
 Finding the number of elements in P x Q The number of elements in $P ", 
 
 Finding the number of elements in P The number of elements in P is n ( P ) = 2 since P = { 4 , 5 } . 
 
 Finding the number of elements in Q The number of elements in Q is n ( Q ) = 3 since Q = { 5 , 6 , 7 } . 
 
 Verifying the formula Now we verify that $n(P ", 
 
 Final Answer Therefore, P = { 4 , 5 } and Q = { 5 , 6 , 7 } . Also, $n(P ",

Examples 
 Understanding Cartesian products is essential in computer science, especially in database design. For instance, if you have a table of customers (P) and a table of products (Q), the Cartesian product P x Q would generate all possible combinations of customers and products. This is a foundational concept for creating relationships between different datasets and performing complex queries.

If $P \times Q =\{(4,5),(4,6),(4,7),(5,5),(5,6),(5,7)\}$, find P and Q . Also show that $n ( P \times Q )= n ( P ) \times n ( Q )$.

Answer (1)

If $P \times Q =\{(4,5),(4,6),(4,7),(5,5),(5,6),(5,7)\}$, find P and Q . Also show that $n ( P \times Q )= n ( P ) \times n ( Q )$.

Answer (1)

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