02 Relations and Functions
Exercise 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:
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Rearrange the equation to isolate the variable: 3x + 1 = 35
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Subtract 1 from both sides: 3x = 34
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Divide both sides by 3: x = 11
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Substitute x = 11 into the original equation: 3(11) + 1 = 35
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Simplify: 32 = 35
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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:
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The set A has 3 elements, so A = {a1, a2, a3}.
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The set B={3,4,5}, so B = {b1, b2, b3}.
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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 NCERT Solutions (Mathematics)
01 Sets
02 Relations and Functions
03 Trigonometric Functions
04 Principle of Mathematical Induction
05 Complex Numbers and Quadratic Equations
06 Linear Inequalities
07 Permutations and Combinations
08 Binomial Theorem
09 Sequences and Series
10 Straight Lines Exercise
10 Straight Lines Miscellaneous
11 Conic Sections
12 Introduction to Three Dimensional Geometry
13 Limits and Derivatives
14 Mathematical Reasoning
15 Statistics
16 Probability