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Q10  The Cartesian product $A \times A$ has 9 elements among which are found (–1, 0) and
(0,1). Find the set A and the remaining elements of $A \times A$

It is given that Cartesian product A × A having 9 elements among which are found (–1, 0) and (0,1). Now, Number of elements in (A× B) = (Number of elements in set A) × (Number of elements in B) It is given that   Therefore,   Now, By definition A × A = {(a, a): a ? A} Therefore, -1, 0 and 1 are the elements of set A  Now, because, n(A) = 3 therefore, A = {-1, 0, 1} Therefore, the...

Q9  Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1)
are in $A \times B$, find A and B, where x, y and z are distinct elements.

It is given that  n(A) = 3 and n(B) = 2 and If (x, 1), (y, 2), (z, 1) are in A × B. By definition of Cartesian product of two non-empty Set P and Q:   Now, we can see that P = set of all first elements. And  Q = set of all second elements. Now,  (x, y, z) are elements of A and (1,2) are elements of B  As n(A) = 3 and n(B) = 2 Therefore, A = {x, y, z} and B = {1, 2}       .   .

Q8   Let A = {1, 2} and B = {3, 4}. Write $A \times B$. How many subsets will $A \times B$ have?
List them.

It is given that A = {1, 2} and B = {3, 4} Then, Now, we know that if C is a set with    Then, Therefore, The set   has    subsets.

Q7 (b)   Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that
$A \times C$ is a subset of $B \times D$

It is given that A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8} Now, And  We can clearly observe that all the elements of the set  are the elements of the set   Therefore,  is a subset  of

Q7 (a)  Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that
$A \times ( B \cap C ) = ( A \times B ) \cap ( A \times C )$.

It is given that A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8} Now, Now, And Now, From equation (i) and (ii) it is clear that Hence,

Q6   If $A \times B$  = {(a, x),(a , y), (b, x), (b, y)}. Find A and B.

It is given that   = {(a, x),(a , y), (b, x), (b, y)} We know that the cartesian product of two non-empty set P and Q is defined as   Now, we know that A is the set of all first elements and B is the set of all second elements Therefore,

Q4 (c)   State whether each of the following statements are true or false. If the statement
is false, rewrite the given statement correctly.

If A = {1, 2}, B = {3, 4}, then$A \times (B \cap \phi ) = \phi$

This statement is  TRUE     If A = {1, 2}, B = {3, 4}, then There for

Q4(2)  State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.  If A and B are non-empty sets, then A × B is a non-empty set of ordered
pairs (x, y) such that $x \epsilon A$ and $y \epsilon B$

It is a  TRUE  statement      If A and B are non-empty sets, then A × B is a non-empty set of ordered  pairs (x, y) such that  and

Q4 (1)   State whether each of the following statements are true or false. If the statement
is false, rewrite the given statement correctly.

If P = {m, n} and Q = { n, m}, then $P \times Q$ = {(m, n),(n, m)}.

FALSE If  P = {m, n} and Q = { n, m}  Then,

Q3  If G = {7, 8} and H = {5, 4, 2}, find  $G \times H \: \: and \: \: H \times G$

It is given that G = {7, 8} and H = {5, 4, 2} We know that the cartesian product of two non-empty sets P and Q is defined as P  Q = {(p,q) , where p  P , q  Q } Therefore, G  H = {(7,5),(7,4),(7,2),(8,5),(8,4),(8,2)} And  H  G = {(5,7),(5,8),(4,7),(4,8),(2,7),(2,8)}

Q2  If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of
elements in $( A \times B )$.

It is given that set A has 3 elements and the elements in set B are 3 , 4 , and 5 Therefore, the number of elements in set B is 3 Now, Number of elements in                                       = ( Number of elements in set A )  ( Number of elements in set B)                                      = 3  3                                       = 9 Therefore, number of elements in  is 9

Q1   If  $\left ( \frac{x}{3}+1 , y - \frac{2}{3} \right ) = \left ( \frac{5}{3},\frac{1}{3} \right )$ , find the values of x and y.

It is given that    Since the ordered pairs are equal, the corresponding elements will also be equal Therefore, Therefore, values of x and y are  2  and 1  respectively
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