[NCERT (average) LEVEL]
Show that the relation are R on the set R 𝑅 = { 𝑥, 𝑦 : 3𝑥 − 𝑦 = 0} is reflexive , symm.
of real numbers as 𝑅 = { 𝑎, 𝑏 : 𝑎 ≤ 𝑏 2 } is or transitive
neither reflexive nor symmetric nor Show that 𝑓: 𝑁 → 𝑁, given by
transitive 𝑓 𝑥 = 𝑥 + 1 , 𝑖𝑓 𝑥 𝑖𝑠 𝑜𝑑𝑑
Check whether the relation R in real 𝑥 − 1 , 𝑖𝑓 𝑥 𝑖𝑠 𝑒𝑣𝑒𝑛
number R defined by 𝑅 = { 𝑎, 𝑏 : 𝑎 ≤ 𝑏 3 } is is bijective.
reflexive , symmetric or transitive Show that 𝑓: 𝑁 → 𝑁, given by
𝑛+1
𝑓 𝑥 = , 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑
2
Show that the relation R in the set 𝑛/2 , 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
𝐴 = {1,2,3,4,5} given by is bijective.
𝑅 = { 𝑎, 𝑏 : 𝑎 − 𝑏 𝑖𝑠 𝑒𝑣𝑒𝑛 } is equivalence Show that function 𝑓: 𝑅 − {3} → 𝑅 − {1}
relation. Show that all elements of {1,3,5} 𝑥−2
given by 𝑓 𝑥 = 𝑥−3 is one-one and onto.
are related to each other and all the
elements of {2,4] are related to each
other but not element of {1,3,5] is Consider 𝑓: 𝑅+ → [4, ∞ given by
related to any element of {2,4} 𝑓 𝑥 = 𝑥 2 + 4, show that f is bijective.
Show that each of the relation in the set Consider 𝑓: 𝑅+ → [−5, ∞ given by
𝐴 = {𝑥 𝜖 𝑧: 0 ≤ 𝑥 ≤ 12} given by 𝑓 𝑥 = 9𝑥 2 + 6𝑥 − 5, show that f is one-
𝑖 𝑅 = { 𝑎, 𝑏 : 𝑎 − 𝑏 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 4} one onto.
𝑖𝑖 𝑅 = { 𝑎, 𝑏 : 𝑎 = 𝑏} Let 𝑓: 𝑁 → 𝑅 be a function defined as
𝑖𝑖𝑖 𝑅 = { 𝑎, 𝑏 : 𝑎 − 𝑏 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 4} 𝑓 𝑥 = 4𝑥 2 + 12𝑥 + 15, Check the function
is equivalence relation. Also Find the one-one and onto.
equivalence class of 1 in each case If 𝑓: 𝑅 → 𝑅 is the function defined by
Ans. (i) [1]= {1,5,9] (ii) [1]= {1} 𝑓 𝑥 = 𝑥 3 + 7, then show that f is a
(iii) [1]= {1,5,9} one-one onto.
Check whether the relation R in the set Let 𝑓: 𝑅 − {4/3} → 𝑅 be a function
4𝑥
𝐴 = {1,2,3,4,5,6} defined as 𝑅 = { 𝑎, 𝑏 : 𝑏 𝑖𝑠 defined as 𝑓 𝑥 = 3𝑥+4 Check function
𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑎} is reflexive , symmetric or one-one and onto.
transitive. Show that function 𝑓: 𝑅 → −1, 1 defined
Check whether the relation R in the set Z 𝑥
by 𝑓 𝑥 = 1+ 𝑥 , 𝑥 ∈ 𝑅 is one-one and
of all integers defined as 𝑅 = { 𝑥, 𝑦 : 𝑥 −
𝑦 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟} is reflexive , symmetric or onto.
transitive. Check the injectivity and surjectivity
Show that the relation R in R defined as 𝑖 𝑓: 𝑁 → 𝑁 , 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = 𝑥 2
𝑅 = { 𝑎, 𝑏 : 𝑎 ≤ 𝑏} is reflexive and transitive 𝑖𝑖 𝑓: 𝑅 → 𝑅 , 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = 𝑥 2
but not symmetric. 𝑖𝑖𝑖 𝑓: 𝑍 → 𝑍 , 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = 𝑥 3
1
Check whether the relation R defined in 𝑖𝑣 𝑓: 𝑅 − {0} → 𝑅 − {0}, 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = 𝑥
the set {1, 2, 3, 4, 5, 6} as 𝑅 = { 𝑎, 𝑏 : 𝑣 𝑓: 𝑅 → 𝑅 , 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = 𝑥
𝑏 = 𝑎 + 1} is reflexive, symmetric or 𝑣𝑖 𝑓: 𝑅 → 𝑅, 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = [𝑥]
transitive. 𝑣𝑖𝑖 𝑓: 𝑅 → 𝑅 , 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = 𝑠𝑔𝑛 𝑥
Show that the relation R in the set Z of 𝑣𝑖𝑖𝑖 𝑓: [0, 𝜋] → 𝑅 , 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = sin 𝑥
integer given by 𝑅 = { 𝑎, 𝑏 : 2 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑎 − 𝑏} If 𝐴 = {1,2,3}, 𝐵 = {4,5,6,7} and let 𝑓 =
is equivalence relation. Write the { 1,4 , 2,5 , 3,6 } be a function from A to
equivalence class [0]. B check one-one onto
Check whether the relation R in the set A relation 𝑅 = { 𝑙1 , 𝑙2 : 𝑙1 ⊥ 𝑙2 }, Check
𝐴 = {1,2,3, … … . .13,14} defined as reflexivity symmetry and transitivity.
Show that the relation are R on the set R 𝑅 = { 𝑥, 𝑦 : 3𝑥 − 𝑦 = 0} is reflexive , symm.
of real numbers as 𝑅 = { 𝑎, 𝑏 : 𝑎 ≤ 𝑏 2 } is or transitive
neither reflexive nor symmetric nor Show that 𝑓: 𝑁 → 𝑁, given by
transitive 𝑓 𝑥 = 𝑥 + 1 , 𝑖𝑓 𝑥 𝑖𝑠 𝑜𝑑𝑑
Check whether the relation R in real 𝑥 − 1 , 𝑖𝑓 𝑥 𝑖𝑠 𝑒𝑣𝑒𝑛
number R defined by 𝑅 = { 𝑎, 𝑏 : 𝑎 ≤ 𝑏 3 } is is bijective.
reflexive , symmetric or transitive Show that 𝑓: 𝑁 → 𝑁, given by
𝑛+1
𝑓 𝑥 = , 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑
2
Show that the relation R in the set 𝑛/2 , 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
𝐴 = {1,2,3,4,5} given by is bijective.
𝑅 = { 𝑎, 𝑏 : 𝑎 − 𝑏 𝑖𝑠 𝑒𝑣𝑒𝑛 } is equivalence Show that function 𝑓: 𝑅 − {3} → 𝑅 − {1}
relation. Show that all elements of {1,3,5} 𝑥−2
given by 𝑓 𝑥 = 𝑥−3 is one-one and onto.
are related to each other and all the
elements of {2,4] are related to each
other but not element of {1,3,5] is Consider 𝑓: 𝑅+ → [4, ∞ given by
related to any element of {2,4} 𝑓 𝑥 = 𝑥 2 + 4, show that f is bijective.
Show that each of the relation in the set Consider 𝑓: 𝑅+ → [−5, ∞ given by
𝐴 = {𝑥 𝜖 𝑧: 0 ≤ 𝑥 ≤ 12} given by 𝑓 𝑥 = 9𝑥 2 + 6𝑥 − 5, show that f is one-
𝑖 𝑅 = { 𝑎, 𝑏 : 𝑎 − 𝑏 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 4} one onto.
𝑖𝑖 𝑅 = { 𝑎, 𝑏 : 𝑎 = 𝑏} Let 𝑓: 𝑁 → 𝑅 be a function defined as
𝑖𝑖𝑖 𝑅 = { 𝑎, 𝑏 : 𝑎 − 𝑏 𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 4} 𝑓 𝑥 = 4𝑥 2 + 12𝑥 + 15, Check the function
is equivalence relation. Also Find the one-one and onto.
equivalence class of 1 in each case If 𝑓: 𝑅 → 𝑅 is the function defined by
Ans. (i) [1]= {1,5,9] (ii) [1]= {1} 𝑓 𝑥 = 𝑥 3 + 7, then show that f is a
(iii) [1]= {1,5,9} one-one onto.
Check whether the relation R in the set Let 𝑓: 𝑅 − {4/3} → 𝑅 be a function
4𝑥
𝐴 = {1,2,3,4,5,6} defined as 𝑅 = { 𝑎, 𝑏 : 𝑏 𝑖𝑠 defined as 𝑓 𝑥 = 3𝑥+4 Check function
𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑎} is reflexive , symmetric or one-one and onto.
transitive. Show that function 𝑓: 𝑅 → −1, 1 defined
Check whether the relation R in the set Z 𝑥
by 𝑓 𝑥 = 1+ 𝑥 , 𝑥 ∈ 𝑅 is one-one and
of all integers defined as 𝑅 = { 𝑥, 𝑦 : 𝑥 −
𝑦 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟} is reflexive , symmetric or onto.
transitive. Check the injectivity and surjectivity
Show that the relation R in R defined as 𝑖 𝑓: 𝑁 → 𝑁 , 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = 𝑥 2
𝑅 = { 𝑎, 𝑏 : 𝑎 ≤ 𝑏} is reflexive and transitive 𝑖𝑖 𝑓: 𝑅 → 𝑅 , 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = 𝑥 2
but not symmetric. 𝑖𝑖𝑖 𝑓: 𝑍 → 𝑍 , 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = 𝑥 3
1
Check whether the relation R defined in 𝑖𝑣 𝑓: 𝑅 − {0} → 𝑅 − {0}, 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = 𝑥
the set {1, 2, 3, 4, 5, 6} as 𝑅 = { 𝑎, 𝑏 : 𝑣 𝑓: 𝑅 → 𝑅 , 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = 𝑥
𝑏 = 𝑎 + 1} is reflexive, symmetric or 𝑣𝑖 𝑓: 𝑅 → 𝑅, 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = [𝑥]
transitive. 𝑣𝑖𝑖 𝑓: 𝑅 → 𝑅 , 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = 𝑠𝑔𝑛 𝑥
Show that the relation R in the set Z of 𝑣𝑖𝑖𝑖 𝑓: [0, 𝜋] → 𝑅 , 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓 𝑥 = sin 𝑥
integer given by 𝑅 = { 𝑎, 𝑏 : 2 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑎 − 𝑏} If 𝐴 = {1,2,3}, 𝐵 = {4,5,6,7} and let 𝑓 =
is equivalence relation. Write the { 1,4 , 2,5 , 3,6 } be a function from A to
equivalence class [0]. B check one-one onto
Check whether the relation R in the set A relation 𝑅 = { 𝑙1 , 𝑙2 : 𝑙1 ⊥ 𝑙2 }, Check
𝐴 = {1,2,3, … … . .13,14} defined as reflexivity symmetry and transitivity.
