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 nonempty sets then A×B is a nonempty 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}