Ch.I. Relations and Functions
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Let ∗ be a ‘binary’ operation on N given by a∗b =LCM (a , b) for all a ,b 𝜖 N. Find 5 ∗ 7.
The binary operation *: 𝑅 × 𝑅 → 𝑅 is defined as 𝑎 ∗ 𝑏 =2a+b. Find (2 ∗ 3) ∗ 4.
Let ∗ be a binary operation on set of integers I, defined by 𝑎 ∗ 𝑏 = 2𝑎 + 𝑏 − 3. Find 3∗ 4.
If ∗ be a binary operation on set of integers I, defined by 𝑎 ∗ 𝑏 = 3𝑎 + 4𝑏 − 2, find 4 ∗ 5.
State the reason for the relation R in the set {1,2,3} given by R={(1,2), (2,1)} not be tranative.
Let 𝐴 = {1,2 ,3}, 𝐵 = {4,5,6,7} and let 𝑓 = {(1,4), (2,5), (3,6)}be a function from A to B. State
whether f is one – one or not.
Given an example to show that the relation R in the set of natural numbers, defined by
𝑅 = {(𝑥, 𝑦), 𝑥, 𝑦 ∈ 𝑁 , 𝑥 ≤ 𝑦 2 } is not transitive.
Write the number of all one – one functions from the set 𝐴 = {𝑎, 𝑏 , 𝑐} to itself.
If 𝑓: 𝑅 → 𝑅 is defined by 𝑓(𝑥) = 3𝑥 + 2, find 𝑓(𝑓(𝑥)).
If the function 𝑓: 𝑅 → 𝑅, defined by 𝑓(𝑥) = 3𝑥 − 4 is invertible, find 𝑓 −1 .
11. What is the range of the function 𝑓(𝑥) =
12. If 𝑓: 𝑅 → 𝑅 be defined by 𝑓(𝑥) = (3 − 𝑥
|𝑥−1|
𝑥−1
3 )1⁄3
.
, then find 𝑓𝑜𝑓(𝑥).
13. Let ∗ be a binary operation on set Q, of rational numbers defined as 𝑎 ∗ 𝑏 =
𝑎𝑏
.
5
Write the Identity
for ∗, if any.
14. If 𝑓: 𝑅 → 𝑅, defined by 𝑓(𝑥) =
3𝑥+5
2
is an invertible function, find 𝑓 −1 .
15. Let ∗ be a binary operation on N given by 𝑎 ∗ 𝑏 = 𝐻. 𝐶. 𝐹. (𝑎, 𝑏), 𝑎, 𝑏 ∈ 𝑁, find the value of 22 ∗ 4.
16. If 𝑓(𝑥) = 𝑥 + 7 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 − 7, 𝑥 ∈ 𝑅, find (𝑓𝑜𝑔)(7).
𝑥 + 1, 𝑖𝑓 𝑥 𝑖𝑠 𝑜𝑑𝑑
17. Show that 𝑓: 𝑁 → 𝑁 given by 𝑓(𝑥) = {
is both one-one and onto.
𝑥 − 1, 𝑖𝑓 𝑥 𝑖𝑠 𝑒𝑣𝑒𝑛
18. Consider the binary operations ∗: 𝑅 × 𝑅 → 𝑅 o: 𝑅 × 𝑅 → 𝑅 be defined 𝑎 ∗ 𝑏 = |𝑎 − 𝑏| and
𝑎𝑜𝑏 = 𝑎, ∀ 𝑎, 𝑏 ∈ 𝑅. Show that ‘*’ is commutative but not associative, ‘o’ is associative but not
commutative.
19. If 𝑓: 𝑅 → 𝑅, be the function defined by 𝑓(𝑥) = 4𝑥 3 + 7, show that f is a bijection.
𝑛 + 1, 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
20. Show that 𝑓: 𝑊 → 𝑊 given by 𝑓(𝑛) = {
is a bijection.
𝑛 − 1, 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑
𝑛 + 1, 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
21. Let 𝑓: 𝑊 → 𝑊 be defined by (𝑛) = {
. Show that 𝑓 is invertible. Find the
𝑛 − 1, 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑
inverse of 𝑓.
𝑥
22. Show that the function 𝑓: 𝑅 → {𝑥 ∈ 𝑅: −1 < 𝑥 < 1} defined by 𝑓(𝑥) = 1+|𝑥| , 𝑥 ∈ 𝑅 is one- one
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and onto function.
Let 𝑓: 𝑅 → 𝑅 be defined by 𝑓(𝑥) = 10𝑥 + 7. Find the function 𝑔: 𝑅 → 𝑅 such that
𝑔𝑜𝑓 = 𝑓𝑜𝑔 = 𝐼𝑅 .
Let N be the set of all natural numbers and R be the relation on NXN, be defined by (a, b)R(c, d) if
ad = bc. Show that R is an Equivalence relation.
Show that the function 𝑓: 𝑅 → 𝑅 be defined by 𝑓(𝑥) = 2𝑥 3 − 7, is a bijection.
Show that the relation is R in the set 𝐴 = {𝑥: 𝑥 ∈ 𝑍, 0 ≤ 𝑥 ≤ 12} given by 𝑅 = {(𝑎, 𝑏) ∶
|𝑎 − 𝑏| 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 4}an equivalence relation. Find set of all elements related to 1.
Show that the function 𝑓: 𝑅 → 𝑅 given 𝑓(𝑥) = 𝑎𝑥 + 𝑏, where 𝑎, 𝑏 𝜖 𝑅, 𝑎 ≠ 0 is a bijection.
Let 𝑓: 𝑋 → 𝑌 be a function. Define a relation 𝑅 on 𝑋 given by 𝑅 = {(𝑎, 𝑏): 𝑓(𝑎) = 𝑓(𝑏)}
Show that 𝑅 is an equivalence relation on 𝑋.
Let 𝑍 be the set of all integers and 𝑅 be the relation on 𝑍 defined as
𝑅 = {(𝑎, 𝑏) ∶ 𝑎, 𝑏 ∈ 𝑍, 𝑎𝑛𝑑 (𝑎 − 𝑏)𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 𝑛}. prove that 𝑅 is an equivalence relation.
3𝑎𝑏
Let * be a binary operation on Q defined by 𝑎 ∗ 𝑏 = 5 .Show that * is commutative as well as
Associative. Also find its identity element, if it exists.
Show that the relation 𝑆: 𝑁 × 𝑁 → 𝑁 is defined by (𝑎, 𝑏) 𝑆 (𝑐, 𝑑) ⇨ 𝑎 + 𝑑 = 𝑏 + 𝑐 is an
equivalence relation.
𝑥+3
If the function 𝑓: 𝑅 → 𝑅 is given by 𝑓(𝑥) = 2 , and 𝑔: 𝑅 → 𝑅 is given by 𝑔(𝑥) = 2𝑥 − 3, find
(i) fog and (ii) gof. Is 𝑓 −1 = g?
33. If the function 𝑓: 𝑅 → 𝑅 is given by 𝑓(𝑥) = 𝑥 2 + 3𝑥 + 1, and 𝑔: 𝑅 → 𝑅 is given by 𝑔(𝑥) = 2𝑥 −
3, find (i) fog and (ii) gof.
𝑎+𝑏
34. (i) Is the binary operation *, defined on set 𝑁, given by 𝑎 ∗ 𝑏 = 2 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑎, 𝑏 ∈ 𝑁,
commutative?
(ii) Is the above binary operation * associative?