1 RELATIONS AND FUNCTIONS EACH OF THE FOLLOWING QUESTIONS CARRIES 1 MARK: Q 1) Define bijective function. A function : → is said to be a bijective function if it is both one one and onto function. Q 2) Define binary operation on a set. An operation ∗ defined on a set is said to be a binary operation if ∀ , ∈ , ∗∈ . Q 3) Define equivalence relation. A relation defined on a set is said to be an equivalence relation if it is reflexive, symmetric and transitive. Q 4) Operation ∗ is defined by ∗ = . Is ∗ a binary operation on + Solution: ∀ , ∈ + , ∗ = ∈ + ∴ ∗ is a binary operation on + . Q 5) Verify whether the operation ∗ defined on by ∗ = +1 is binary or not. Solution: ∀ , ∈ , ∗ = +1 ∈ ∴ ∗ is a binary operation on . Q 6) Let ∗ be the binary operation on given by ∗ = of and , find 20 ∗ 16 . Solution: 20 ∗ 16 = of 20 and 16 = 80 . EACH OF THE FOLLOWING QUESTIONS CARRIES 2 MARKS: Q 1) A binary operation ^ on the set {1, 2, 3, 4, 5} is defined by ^= min{, }. Write the operation table for the operation ^. Solution: ^ 1 2 3 4 5 1 1 1 1 1 1 2 1 2 2 2 2 3 1 2 3 3 3 4 1 2 3 4 4 5 1 2 3 4 5 Q 2) Verify whether the operation ∗ defined on by ∗ = 4 is associative or not. Solution: ∗ ∗ = 4 ∗ = 16 ∗ (∗)= ∗ 4 = 16 ∗ ∗ = ∗ (∗ ) ∴ ∗ is associative on .
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RELATIONS AND FUNCTIONS
EACH OF THE FOLLOWING QUESTIONS CARRIES 1 MARK:
Q 1) Define bijective function.
A function 𝑓: 𝐴 → 𝐵 is said to be a bijective function if it is both one one and onto function.
Q 2) Define binary operation on a set.
An operation ∗ defined on a set 𝐴 is said to be a binary operation if ∀ 𝑎, 𝑏 ∈ 𝐴 , 𝑎 ∗ 𝑏 ∈ 𝐴 .
Q 3) Define equivalence relation.
A relation 𝑅 defined on a set 𝐴 is said to be an equivalence relation if it is reflexive, symmetric
and transitive.
Q 4) Operation ∗ is defined by 𝑎 ∗ 𝑏 = 𝑎 . Is ∗ a binary operation on 𝑍+
Solution: ∀ 𝑎, 𝑏 ∈ 𝑍+ , 𝑎 ∗ 𝑏 = 𝑎 ∈ 𝑍+
∴ ∗ is a binary operation on 𝑍+ .
Q 5) Verify whether the operation ∗ defined on 𝑍 by 𝑎 ∗ 𝑏 = 𝑎𝑏 + 1 is binary or not.
Solution: ∀ 𝑎, 𝑏 ∈ 𝑍, 𝑎 ∗ 𝑏 = 𝑎𝑏 + 1 ∈ 𝑍
∴ ∗ is a binary operation on 𝑍 .
Q 6) Let ∗ be the binary operation on 𝑁 given by 𝑎 ∗ 𝑏 = 𝐿𝐶𝑀 of 𝑎 and 𝑏 , find 20 ∗ 16 .
Solution: 20 ∗ 16 = 𝐿𝐶𝑀 of 20 and 16 = 80 .
EACH OF THE FOLLOWING QUESTIONS CARRIES 2 MARKS:
Q 1) A binary operation ^ on the set {1, 2, 3, 4, 5} is defined by 𝑎^𝑏 = min{𝑎, 𝑏}.
Write the operation table for the operation ^.
Solution:
^ 1 2 3 4 5
1 1 1 1 1 1
2 1 2 2 2 2
3 1 2 3 3 3
4 1 2 3 4 4
5 1 2 3 4 5
Q 2) Verify whether the operation ∗ defined on 𝑄 by 𝑎 ∗ 𝑏 =𝑎𝑏
4 is associative or not.
Solution: 𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎𝑏
4 ∗ 𝑐 =
𝑎𝑏𝑐
16
𝑎 ∗ (𝑏 ∗ 𝑐) = 𝑎 ∗ 𝑏𝑐
4 =
𝑎𝑏𝑐
16
𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎 ∗ (𝑏 ∗ 𝑐) ∴ ∗ is associative on 𝑄 .
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Q 3) Let ∗ be the binary operation on 𝑄 given by 𝑎 ∗ 𝑏 =𝑎𝑏
4 . Find the identity element.
Solution: 𝑎 ∗ 𝑒 = 𝑎 = 𝑒 ∗ 𝑎
𝑎 ∗ 𝑒 = 𝑎 ∴𝑎𝑒
4= 𝑎 ⟹ 𝑒 = 4
EACH OF THE FOLLOWING QUESTIONS CARRIES 3 MARKS:
Q 1) Show that the relation 𝑅 in the set of integers given by 𝑅 = 𝑎, 𝑏 : 5 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑎 − 𝑏 is an
equivalence relation.
Solution: 𝒂 − 𝒂 = 𝟎
5 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 0
⟹ 5 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑎 − 𝑎
∴ ∀ 𝑎 ∈ 𝑍, (𝑎, 𝑎) ∈ 𝑅
∴ 𝑅 is a reflexive relation.
Let 𝑎, 𝑏 ∈ 𝑅 ⟹ 5 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑎 − 𝑏
⟹ 5 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 − 𝑎 − 𝑏
⟹ 5 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑏 − 𝑎
⟹ 𝑏, 𝑎 ∈ 𝑅
∴ 𝑅 is a symmetric relation.
Let 𝑎, 𝑏 ∈ 𝑅 𝑎𝑛𝑑 𝑏, 𝑐 ∈ 𝑅
⟹ 5 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑎 − 𝑏 𝑎𝑛𝑑 5 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑏 − 𝑐
⟹ 5 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 (𝑎 − 𝑏 + 𝑏 − 𝑐)
⟹ 5 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑎 − 𝑐
⟹ 𝑎, 𝑐 ∈ 𝑅
∴ 𝑅 is a transitive relation.
Since 𝑅 is reflexive, symmetric and transitive, it is an equivalence relation.
Q 2) Determine whether the relation 𝑅 in the set 𝐴 = 1, 2, 3, 4, 5, 6 as
𝑅 = { 𝑥, 𝑦 : 𝑦 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 𝑥} is an equivalence relation.
Solution: 𝑥 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 𝑥
∴ ∀ 𝑥 ∈ 𝐴, (𝑥, 𝑥) ∈ 𝑅
∴ 𝑅 is a reflexive relation.
Now 2, 4 ∈ 𝑅
⟹ 4 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 2
But 2 𝑖𝑠 𝑛𝑜𝑡 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 4
⟹ (4, 2) ∉ 𝑅
∴ 𝑅 is not a symmetric relation.
Let 𝑥, 𝑦 ∈ 𝑅 𝑎𝑛𝑑 𝑦, 𝑧 ∈ 𝑅
⟹ 𝑦 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 𝑥 𝑎𝑛𝑑 𝑧 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 𝑦
⟹ 𝑧 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 𝑥
⟹ (𝑧, 𝑥) ∈ 𝑅
∴ 𝑅 is a transitive relation.
Since 𝑅 is reflexive and transitive but not symmetric it is not an equivalence relation.
Q 3) If ∗ is a binary operation defined on 𝐴 = 𝑁 × 𝑁 by 𝑎, 𝑏 ∗ 𝑐, 𝑑 = 𝑎 + 𝑐, 𝑏 + 𝑑 , prove
that ∗ is both commutative and associative. Find the identity, if it exists.
Solution: 𝑎, 𝑏 ∗ 𝑐, 𝑑 = 𝑎 + 𝑐, 𝑏 + 𝑑
3
𝑐, 𝑑 ∗ 𝑎, 𝑏 = 𝑐 + 𝑎, 𝑑 + 𝑏
∀ 𝑎, 𝑏 , 𝑐, 𝑑 ∈ 𝐴 , 𝑎, 𝑏 ∗ 𝑐, 𝑑 = 𝑐, 𝑑 ∗ 𝑎, 𝑏
∴ ∗ is commutative.
𝑎, 𝑏 ∗ 𝑐, 𝑑 ∗ , 𝑘 = 𝑎 + 𝑐, 𝑏 + 𝑑 ∗ , 𝑘
= 𝑎 + 𝑐 + , 𝑏 + 𝑑 + 𝑘
𝑎, 𝑏 ∗ 𝑐, 𝑑 ∗ , 𝑘 = 𝑎 ∗ 𝑏 ∗ 𝑐 + , 𝑑 + 𝑘
= 𝑎 + 𝑐 + , 𝑏 + 𝑑 + 𝑘
𝑎, 𝑏 ∗ 𝑐, 𝑑 ∗ , 𝑘 = 𝑎, 𝑏 ∗ 𝑐, 𝑑 ∗ , 𝑘
∴ ∗ is associative.
If possible let 𝑒, 𝑒′ be the identity element.
∴ 𝑎, 𝑏 ∗ 𝑒, 𝑒′ = 𝑎, 𝑏 = 𝑒, 𝑒′ ∗ 𝑎, 𝑏
⟹ 𝑎 + 𝑒, 𝑏 + 𝑒′ = 𝑎
⟹ 𝑎 + 𝑒 = 𝑎
⟹ 𝑒 = 0 ∉ 𝑁
The identity element does not exist.
EACH OF THE FOLLOWING QUESTIONS CARRIES 5 MARKS:
Q 1) Verify whether the function 𝑓: 𝑁 → 𝑌 defined by 𝑓 𝑥 = 4𝑥 + 3 where
𝑌 = 𝑦: 𝑦 = 4𝑥 + 3, 𝑥 ∈ 𝑁 is invertible or not. Write the inverse of 𝑓 𝑥 if it exists.
Solution: Let 𝑓 𝑥1 = 𝑓(𝑥2) where 𝑥1, 𝑥2 ∈ 𝑁
4𝑥1 + 3 = 4𝑥2 + 3
⟹ 4𝑥1 = 4𝑥2
⟹ 𝑥1 = 𝑥2
∴ 𝑓 𝑥 is a one one function.
Let 𝑦 = 𝑓 𝑥
⟹ y = 4x + 3
⟹ 𝑥 =𝑦−3
4∈ 𝑁
For every 𝑦 ∈ 𝑌 there exists an 𝑥 =𝑦−3
4∈ 𝑁 such that 𝑓 𝑥 = 𝑦.
∴ 𝑓 𝑥 is an onto function.
∵ 𝑓 𝑥 is both one one and onto, it is invertible.
The inverse of 𝑓 𝑥 exists and is given by 𝑓−1 𝑦 =𝑦−3
4 .
Q 2) If 𝑅+ is the set of all non negative real numbers, prove that the function 𝑓: 𝑅+ → 4, ∞
defined by 𝑓 𝑥 = 𝑥2 + 4 is invertible. Also write the inverse of 𝑓 𝑥 .
Solution: Let 𝑓 𝑥1 = 𝑓(𝑥2) where 𝑥1, 𝑥2 ∈ 𝑁
⟹ 𝑥12 + 4 = 𝑥2
2 + 4
⟹ 𝑥12 = 𝑥2
2
⟹ 𝑥1 = 𝑥2
∴ 𝑓 𝑥 is a one one function.
Let 𝑦 = 𝑓 𝑥
⟹ y = 𝑥2 + 4
𝑥 = 𝑦 − 4 ∈ 𝑅+
4
For every 𝑦 ∈ 4,∞ there exists an 𝑥 = 𝑦 − 4 ∈ 𝑅+ such that 𝑓 𝑥 = 𝑦.
∴ 𝑓 𝑥 is an onto function.
∵ 𝑓 𝑥 is both one one and onto, it is invertible.
The inverse of 𝑓 𝑥 exists and is given by 𝑓−1 𝑦 = 𝑦 − 4 .
Q 3) Let 𝑓: 𝑁 → 𝑅 be defined by 𝑓 𝑥 = 4𝑥2 + 12𝑥 + 15. Show that 𝑓: 𝑁 → 𝑆, where 𝑆 is the
range of the function is invertible. Also find the inverse of 𝑓.
Solution: Let 𝑓 𝑥1 = 𝑓(𝑥2) where 𝑥1, 𝑥2 ∈ 𝑁
⟹ 4𝑥12 + 12𝑥1 + 15 = 4𝑥2
2 + 12𝑥2 + 15
⟹ 2𝑥1 + 3 2 + 6 = 2𝑥2 + 3 2 + 6
⟹ 2𝑥1 + 3 2 = 2𝑥2 + 3 2
⟹ 2𝑥1 + 3 = 2𝑥2 + 3
⟹ 2𝑥1 = 2𝑥2
⟹ 𝑥1 = 𝑥2
∴ 𝑓 𝑥 is a one one function.
Let 𝑦 = 𝑓 𝑥
⟹ 𝑦 = 4𝑥2 + 12𝑥 + 15
⟹ 2𝑥 + 3 2 + 6 = 𝑦
⟹ 2𝑥 + 3 2 = 𝑦 − 6
⟹ 2𝑥 + 3 = 𝑦 − 6
⟹ 𝑥 = 𝑦−6 − 3
2∈ 𝑁
For every 𝑦 ∈ 𝑆 there exists an 𝑥 = 𝑦−6 − 3
2∈ 𝑁 such that 𝑓 𝑥 = 𝑦.
∴ 𝑓 𝑥 is an onto function.
∵ 𝑓 𝑥 is both one one and onto, it is invertible.
The inverse of 𝑓 𝑥 exists and is given by 𝑓−1 𝑦 = 𝑦−6 − 3
2 .
Q 4) If 𝑅+ is the set of all non negative real numbers prove that the function
𝑓: 𝑅+ → [−5, ∞) defined by 𝑓 𝑥 = 9𝑥2 + 6𝑥 − 5 is invertible. Also find 𝑓−1(𝑥).
Solution: Let 𝑓 𝑥1 = 𝑓(𝑥2) where 𝑥1, 𝑥2 ∈ 𝑅+
⟹ 9𝑥12 + 6𝑥1 − 5 = 9𝑥2
2 + 6𝑥2 − 5
⟹ 3𝑥1 + 1 2 − 6 = 3𝑥2 + 1 2 − 6
⟹ 3𝑥1 + 1 2 = 3𝑥2 + 1 2
⟹ 3𝑥1 + 1 = 3𝑥2 + 1
⟹ 3𝑥1 = 3𝑥2
⟹ 𝑥1 = 𝑥2
∴ 𝑓 𝑥 is a one one function.
Let 𝑦 = 𝑓 𝑥
⟹ 𝑦 = 9𝑥2 + 6𝑥 − 5
⟹ 3𝑥 + 1 2 − 6 = 𝑦
⟹ 3𝑥 + 1 2 = 𝑦 + 6
⟹ 3𝑥 + 1 = 𝑦 + 6
⟹ 𝑥 = 𝑦+6 − 1
3∈ 𝑅+
5
For every 𝑦 ∈ [−5, ∞) there exists an 𝑥 = 𝑦+6 − 1
3∈ 𝑅+ such that 𝑓 𝑥 = 𝑦.
∴ 𝑓 𝑥 is an onto function.
∵ 𝑓 𝑥 is both one one and onto, it is invertible.
The inverse of 𝑓 𝑥 exists and is given by 𝑓−1 𝑦 = 𝑦+6 − 1