CARTESIAN PRODUCT OF SET

1) Express {(x,y): y+ 2x = 5, x, y ∈ W} as the set of ordered pairs.          {(0,5),(1,3),(2,1)}

)  Express {(x,y): y+ 2x = 5, x, y ∈ N} as the set of ordered pairs.       {(1,3),(2,1)}

1) If the ordered pairs (x-1, y+ 3) and (2, x+4) are equal, find x ,y.                  3,4 

) If the ordered pairs (a, -1) and (5, b) belong to {(x,y): y = 2x -3}, find the value of a, b.                   1,7  

) Find a, b if (a/b, b+5)=(-1,-2).        7,-7

) If the ordered pairs (x- 3, 2y+ 3) = (y-2, x+4). find x ,y.                           3,2

) If x ∈ {2,3,5} and y ∈ {2,4,6}, form the set of all ordered pairs (x,y) such that x < y.             {(2,4),+2,6),(3,4),(3,6),(5,6)} 

) Let A={2,3,4} and B {3,5}. Find

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

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

C) A x A.       {(2,2),(2,3),(2,4),(3,2),(3,3),(3,4),(4,2),(4,3),(4,4)}

D) B x B.      {(3,3),(3,5),(5,3),(5,5)}

) Let A={2,3} and B {0,1,5}. Find

A) A x B.      {(2,0),(2,1),(2,5),(3,0),(3,1)(3,5)}

B) A x A.               {(2,2),(2,3),(3,2),(3,3).

) Let A={2,3,4} and B {0,1}. Find

A) A x B.      {(2,0),(2,1),(3,0),(3,1),(4,0),(4,1)}

B) B x A.      {(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)}

C) A x A.        {(2,2),(2,3),(2,4),(3,2),(3,3),(3,4),(4,2),(4,3),(4,4)}

D) B x B.       {(0,0),(0,1),(1,0),(1,1)}

E) n(A x B).                      6

F) n(B xA).              6

G) n(A xA).         9

H) n(B xB).       4

I) Is A x B = B x A ?       No

) Let A={a,b}, B={b,c} and C={b,c,d}.

Verify

A) A x (B U C)= (A x B)U(A xC)

B) A x (B ∩ C) = (A x B) ∩(A x C)

C) (C - A) x B = (C x B) - (A x B)

) Let A={x : x ∈N, 3< x ≤ 6}, B={x : x ∈ N, 2≤ x < 4} and C={x : x∈ N, 3≤ x <6}. Show that

A) (C - B) x A = (C x A)- (B x A).

B)  (C - A) x B = (C x B) - (A x B).

) 

) If A x B={(2,1),(2,4),(3,1),(3,4),(5,1),(5,4)} , find A and B.             A={2,3,5}, B={1,4} 

) Let A and B be two sets such that A x B contains 6 elements. If three elements of A x B are (1,3),(2,5) and (3,3), find its remaining elements.      If A x B={(1,5),(2,3),(3,5)

) Let A={1,2,3} and B {3,4}. Represent the following products by arrow diagram:

A) A x B.    

B) B x A.     

C) A x A.      

D) B x B.     

) State True or False 

a) A x ∅ = ∅

b) n(A) = x, n(B)= y and A ∩ B = ∅, then n(Ax B)= xy.

c) n(A) = x, n(B)= y and (A x A) = x², and n(Bx B)= y².

d) {(2,a),(3,b),(2,b),(3,a),(3,c)} is a Cartesian product set.

e) If A={1,2,3} and B={0,1} then (3,2) ∈ A x B.

f) If P={m,n} and Q={n,m} then P x Q ={(m,n),(n,m)}.

g) If A and B are non-empty sets, then A x B is a non empty set of order pairs (x,y) such that x ∈ B and y ∈ A.

h) {(a,x),(a,y),(b,x),(b,y)} is a product set.

i) {(a,x),(a,y),(a,z),(b,y)} is a product set.

EXERCISE

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