All Questions with Enchanted Animations
Let A = {1, 2, 3, 7} and B = {3, 0, –1, 7}. Which of the following are relations from A to B?
Wizard's Rule: A relation from A to B must have first elements from A and second elements from B.
(i) R₁ = {(2,1), (7,1)}:
• (2,1): 2 ∈ A ✓ but 1 ∉ B ✗
• (7,1): 7 ∈ A ✓ but 1 ∉ B ✗
→ Not a relation
(ii) R₂ = {(–1,1)}:
• (–1,1): –1 ∉ A ✗
→ Not a relation
(iii) R₃ = {(2,–1), (7,7), (1,3)}:
• (2,–1): 2 ∈ A ✓ and –1 ∈ B ✓
• (7,7): 7 ∈ A ✓ and 7 ∈ B ✓
• (1,3): 1 ∈ A ✓ and 3 ∈ B ✓
→ Valid relation
(iv) R₄ = {(7,–1), (0,3), (3,3), (0,7)}:
• (7,–1): Valid
• (0,3): 0 ∉ A ✗
• (3,3): Valid
• (0,7): 0 ∉ A ✗
→ Not a relation
Let A = {1, 2, 3, ..., 45} and R be the relation defined as "is square of" on A.
Write R as a subset of A × A. Also find the domain and range of R.
The relation R contains pairs where the first element is the square of the second.
Possible pairs: (1,1), (4,2), (9,3), (16,4), (25,5), (36,6), (49,7), etc.
But A only goes up to 45, so we exclude pairs where first element > 45:
• 7² = 49 > 45 → exclude
Final R = { (1,1), (4,2), (9,3), (16,4), (25,5), (36,6) }
Domain: {1, 4, 9, 16, 25, 36}
Range: {1, 2, 3, 4, 5, 6}
A Relation R is given by the set { (x, y) | y = x + 3 }, where x ∈ {0,1,2,3,4,5}. Determine its domain and range.
Calculate y for each x using y = x + 3:
| x | y = x + 3 | Pair |
|---|---|---|
| 0 | 3 | (0,3) |
| 1 | 4 | (1,4) |
| 2 | 5 | (2,5) |
| 3 | 6 | (3,6) |
| 4 | 7 | (4,7) |
| 5 | 8 | (5,8) |
Domain: {0, 1, 2, 3, 4, 5}
Range: {3, 4, 5, 6, 7, 8}
Represent each of the given relations by (a) an arrow diagram, (b) a graph and (c) a set in roster form.
(i) { (x,y) | x = 2y, x ∈ {2,3,4,5}, y ∈ {1,2,3,4} }
Roster form:
• y=1 → x=2 → (2,1)
• y=2 → x=4 → (4,2)
• y=3 → x=6 → invalid (6∉A)
• y=4 → x=8 → invalid (8∉A)
R = { (2,1), (4,2) }
2 → 1
4 → 2
Points on x = 2y:
(2,1) and (4,2)
(ii) { (x,y) | y = x + 3, x, y are natural numbers < 10 }
Roster form:
• x=1 → y=4 → (1,4)
• x=2 → y=5 → (2,5)
• x=3 → y=6 → (3,6)
• x=4 → y=7 → (4,7)
• x=5 → y=8 → (5,8)
• x=6 → y=9 → (6,9)
R = { (1,4), (2,5), (3,6), (4,7), (5,8), (6,9) }
1 → 4
2 → 5
3 → 6
4 → 7
5 → 8
6 → 9
Points on y = x + 3
Forms a straight line
A company has four categories of employees:
If the relation R is defined by xRy, where x is the salary given to person y, express R through ordered pairs and an arrow diagram.
Ordered pairs:
For Assistants: (10000, A₁), (10000, A₂), (10000, A₃), (10000, A₄), (10000, A₅)
For Clerks: (25000, C₁), (25000, C₂), (25000, C₃), (25000, C₄)
For Managers: (50000, M₁), (50000, M₂), (50000, M₃)
For Executives: (100000, E₁), (100000, E₂)
10000 → A₁, A₂, A₃, A₄, A₅
25000 → C₁, C₂, C₃, C₄
50000 → M₁, M₂, M₃
100000 → E₁, E₂