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space space space space space space space space open parentheses straight x over 3 plus 1 comma space straight y minus 2 over 3 close parentheses equals open parentheses 5 over 3 comma 1 third close parentheses,  find the values of x and y.


space space space space space space space space space space space space space space space space open parentheses straight x over 3 plus 1 comma space straight y minus 2 over 3 close parentheses equals open parentheses 5 over 3 comma 1 third close parentheses

rightwards double arrow      space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight x over 3 plus 1 equals 5 over 3 space space or space straight x over 3 equals 5 over 3 or space straight x space equals space 2

and           space space space space space space space space space space space space space space space space space space space space space space space space space space space straight y minus 2 over 3 equals 1 third or space straight y equals 1 third plus 2 over 3 equals 3 over 3 or space straight y equals 1

Hence, x =2, y =1
533 Views

State whether each of the following statements are True or False. If the statement is False, re-write it correctly.

 (i) P = {m, n} and Q = {n, m}, then P x Q = {(m, n), (n, m)}
(ii) If A and B are two non-empty sets, then A x B is a non-empty set of ordered pairs (x, y) such that space space space space straight x element of straight B space and space straight y element of straight A.
(iii) If A = {1, 2}, B = {3, 4}, then space space space space space space space space space space straight A space cross times left parenthesis straight B intersection straight ϕ right parenthesis equals straight ϕ

(i)  P = {m, n},  Q = {n, m}

rightwards double arrow  n(P|) = 2,  n(Q) = 2

rightwards double arrow n(P xQ) =n(P) x n(Q) = 2 x 2 = 4

But, it is given that P x Q  = {(m, n).(n, m)}

rightwards double arrown(P x Q) = 2

Hence, the statement is False.

Also,     P x Q = {m, n} x {n , m} = {(m, m), (m, n), (n, n), (n, m)}


(ii) The statement is False. The correct statement is space space space space space space space space space space space straight A cross times straight B equals left curly bracket left parenthesis straight x comma space straight y right parenthesis space colon space straight x element of straight A comma space straight y element of straight B right curly bracket

(iii) The statement is true as space space space space space space space space space space space space straight A cross times left parenthesis straight B intersection straight ϕ right parenthesis space equals space left curly bracket 1 comma space 2 right curly bracket space straight X space left square bracket left curly bracket 3 comma space 4 right curly bracket intersection straight ϕ right square bracket space equals space left curly bracket 1 comma space 2 right curly bracket space cross times straight ϕ space space space space open square brackets space because space straight B intersection straight ϕ equals straight ϕ close square brackets


                                                                                           space space space space space space space space space space space space space space space space space space space space space open square brackets because space space straight A space cross times space straight ϕ close square brackets space equals space straight ϕ


210 Views

If A × B = {(a, x),(a , y), (b, x), (b, y)}. Find A and B.


                      A x B = {(p, q), (p, r), (m, q), (m, r)}

space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left parenthesis straight p comma straight q right parenthesis element of straight A cross times straight B space rightwards double arrow space straight p element of straight A comma straight q element of straight B


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        space space space space space space space space space space space space space space space space space space space left parenthesis straight m comma space straight q right parenthesis element of straight A space cross times space straight B space rightwards double arrow space straight m element of straight A comma space straight q element of straight B

            space space space space space space space space space space space space space left parenthesis straight m comma space straight r right parenthesis element of straight A space cross times space straight B space rightwards double arrow space straight m element of straight A comma space straight r element of straight B


Hence,    A = {p, m), B = {q, r}

189 Views

If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G


G x H = (7, 8) x (5, 4, 2)

= {(7, 5).(7,4),(7,2),(8,5)(8,4),(8,2)}

H x G = {5, 4, 2) x (7, 8}

= {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)}

215 Views

If A = {–1, 1}, find A × A × A.


A X A X A = {(-1, -1, -1) (-1, -1, 1)(-1, 1, -1), (-1, 1, 1), (1, -1, -1), (1, -1: 1), (1, 1, -1), (1, 1, 1)}

202 Views