02 ସମ୍ପର୍କ ଏବଂ କାର୍ଯ୍ୟ
ବ୍ୟାୟାମ 1
Question:
State whether each of the following statements are true or false. If the statement is false rewrite the given statement correctly (i) If P={m,n} and Q={n,m} then P×Q={(m,n),(n,m)} (ii) If A and B are non-empty sets then A×B is a non-empty set of ordered pairs (x,y) such that x∈A and y∈B (iii) If A={1,2},B={3,4} then A×(B∩ϕ)=ϕ
Answer:
(i) True (ii) True (iii) False; A×(B∩ϕ)=ϕ is an empty set
Question:
If A×B={(a,x),(a,y),(b,x),(b,y)} Find A and B.
Answer:
A = {a,b} B = {x,y}
Question:
The cartesian product A×A has 9 elements among which are found (−1,0) and (0,1). Find the set A and the remaining elements of A×A.
Answer:
Set A = {-1, 0, 1}
The remaining elements of A×A are: (1, -1), (1, 0), (1, 1), (0, -1), (0, 0), (-1, 1), (-1, -1), (-1, 0)
Question:
If (3x+1,y−32)=(35,31) find the values of x and y
Answer:
-
Rearrange the equation to isolate the variable: 3x + 1 = 35
-
Subtract 1 from both sides: 3x = 34
-
Divide both sides by 3: x = 11
-
Substitute x = 11 into the original equation: 3(11) + 1 = 35
-
Simplify: 32 = 35
-
Subtract 32 from both sides: y = 31
Question:
Let A={1,2},B={1,2,3,4},C={5,6} and D={5,6,7,8} Verify that (i) A×(B∩C)=(A×B)∩(A×C) (ii) A×C is a subset of B×D
Answer:
(i) A×(B∩C) = { (1,1), (1,2), (2,1), (2,2) } (A×B)∩(A×C) = { (1,1), (1,2), (2,1), (2,2) }
A×(B∩C) = (A×B)∩(A×C)
(ii) A×C = { (1,5), (1,6), (2,5), (2,6) } B×D = { (1,5), (1,6), (1,7), (1,8), (2,5), (2,6), (2,7), (2,8), (3,5), (3,6), (3,7), (3,8), (4,5), (4,6), (4,7), (4,8) }
A×C is a subset of B×D
Question:
If the set A has 3 elements and the set B={3,4,5} then find the number of elements in (A×B)?
Answer:
Answer:
-
The set A has 3 elements, so A = {a1, a2, a3}.
-
The set B={3,4,5}, so B = {b1, b2, b3}.
-
The number of elements in (A×B) = 3 × 3 = 9.
Question:
If A = {-1, 1}, find A × A × A.
Answer:
A × A × 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}
= {-1, -1, -1, 1, -1, 1, -1, 1}
Question:
If A = {-1, 1}, find A × A × A.
Answer:
A × A × 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}
= {-1, -1, -1, 1, -1, 1, -1, 1}
Question:
Let A={1,2} and B={3,4}. Write A×B. How many subsets will A×B have?
Answer:
A x B = {(1,3), (1,4), (2,3), (2,4)}
A x B will have 4 subsets.
Question:
If G={7,8} and H={5,4,2}, find G×H and H×G.
Answer:
G×H = {(7,5), (7,4), (7,2), (8,5), (8,4), (8,2)}
H×G = {(5,7), (5,8), (4,7), (4,8), (2,7), (2,8)}
Question:
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×B find A and B where x,y and z are distinct elements
Answer:
A = {x,y,z} B = {1,2}
JEE ଅଧ୍ୟୟନ ସାମଗ୍ରୀ (ଗଣିତ)
01 ସେଟ୍
02 ସମ୍ପର୍କ ଏବଂ କାର୍ଯ୍ୟ
03 ଟ୍ରାଇଗୋନେଟ୍ରିକ୍ କାର୍ଯ୍ୟଗୁଡ଼ିକ
04 ଗାଣିତିକ ଅନୁକରଣର ନୀତି
05 ଜଟିଳ ସଂଖ୍ୟା ଏବଂ ଚତୁର୍ଭୁଜ ସମୀକରଣ
06 ରେଖା ଅସମାନତା
07 ଅନୁମତି ଏବଂ ମିଶ୍ରଣ
08 ଦ୍ୱିପାକ୍ଷିକ ତତ୍ତ୍।
09 କ୍ରମ ଏବଂ କ୍ରମ
10 ସିଧା ଲାଇନ ବ୍ୟାୟାମ
10 ସିଧା ରେଖା ବିବିଧ
11 କନିକ୍ ବିଭାଗ
12 ତିନୋଟି ଡାଇମେନ୍ସନାଲ୍ ଜ୍ୟାମିତିର ପରିଚୟ
13 ସୀମା ଏବଂ ଡେରିଭେଟିଭ୍
14 ଗାଣିତିକ କାରଣ
15 ପରିସଂଖ୍ୟାନ
16 ସମ୍ଭାବନା