SUCCESS STUDY CIRCLE
TIME-3HRS SUB: MATH (RELATION FUNCTION) RIVISION FM-100
GROUP- A
1. LONG. QUESTION ANSWERS ALL. EACH QUESTION 5 MARK.
(a).Show that (i)ca≡cb (mod m)⟹ a ≡ b (mod m’) where m = m’ x gcd (c,m)
(ii) ca≡cb (mod m)⟹ a ≡ b (mod m) if gcd (c, m) =1
(b)Show that the relation ~ on Z given by ~ = {(a,b): a≡b (mod 3) is an equivalence relation.
(c)Congruence modulo 3 relation partitions (or decomposes ) the set Z into three disjoint equivalence classes.
(d)Let X and Y be finite sets and |𝑋|=|𝑌|. Show that if ƒ : 𝑋 → 𝑌 𝑜𝑠 𝑜𝑛𝑡𝑜 𝑡ℎ𝑒𝑛 ƒ must be one one conversely if ƒ is one-one
then it must be onto.
(e)Show that ƒ: R → Rdefined by ƒ (x ) = x2 – 1is not invertible in general. Find the domain an codomain where ƒ is
invertible. Also find ƒ-1.
(f)Let ƒ: R → Rdefined by ƒ (x ) =3x + 5 show that ƒis bijective. Find ƒ-1(1) and ƒ-1(0).
(g)Let ƒ: : X → Y. If there exists α map g : : Y → X such that goƒ=idx and ƒog =idy, then show that (i) ƒ is bijective and
(ii) g = ƒ-1.
(h)Constract the composition table/multiplication table for the binary operation * defined on {0, 1, 2, 3, 4} by a * b= a x b
(mod 5). Find the identity element if any. Also find the inverse elements of 2 and 4.
(i)Show that functions ƒ and g defined by ƒ(x)=2 log x and g(x) = log x2 are not equal even though log x2 = 2log x.
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𝑥 2 − 𝑥2 , 𝑥 ≠ 0
(j)Let ƒ: R → R be defined by ƒ(x)= 𝑓(𝑥) = { show that ƒ is on to but not one-one.
0 , 𝑥=0
GROUP-B
2.SHORT TYPE QUESTION. ANSWER ALL. EACH QUESTION 3 MARK.
(a)List the members of the equivalence relation defined by the following partition on X={1, 2, 3, 4}. Also find the
equivalence classes of 1, 2, 3 and 4 .(i) {{1}, {2}, {3, 4}} (ii){{1, 2, 3}, {4}} (iii){{1, 2, 3, 4}}
(b)Find the least positive integer r such that (i) 185 ∈ [r]7 (ii)-375∈[r]11 (iii)-12 ∈ [r]13.
(c)Find least positive integer x satisfying 276x + 128 ≡ 4(mod 7)
(d)Consider the following real functions. (i)ƒ 1(x)= sin x2 (ii)ƒ2(x)= cos (sin x) (iii)ƒ3(x)= (2x+1)2 + 3(2x+1)+2 . Hence
ƒ 1, ƒ2 and ƒ3 are some examples of functions obtained by composition of ƒ and g in the following ways (Assuming ƒ: R→ R
and g : R→ R).
(e)Let ƒ= {(1,a), 2, b), (3, c), (4, d)} and g= {(a, x), (b, x), (c, y), (d, x)}
(f)Let ƒ= (1, 3), (2, 4), (3, 7)} and g= {(3, 2) (4, 3), (7, 1)}
(g) Prove that the following sets are equivalent: {1.2,3,4,5,6,....} {2,4,6,8,10,....} {1,7,5,7,9,...} {1, 4,9,16,25}
GROUP- C
3. SHORT TYPE QUESTION. ANSWER ALL. EACH QUESTION 2 MARK.
Determine whether the following operation as defined by * are binary operations on the sets specified in each case. Give
Reasons if it is not a binary operation.
(a)a*b= 2a+3b on Z.
(b)a*b=GCD {a,b}on {0,1,2,3,4....,10}
(c)a*b= √𝑎2 + 𝑏 2 on Q +
(d)a*b=LCM{a,b} on N
(e)a*b=a+b-ab on R-{1}.
4. SHORT TYPE QUESTION. ANSWER ALL. EACH QUESTION 2 MARK.
(a)Let ƒ be a real function. Show that h(x)=ƒ(x) + ƒ(-x) is always an even function and g(x)= ƒ(x) –ƒ(-x) is always an odd
function.
(b)Let X= {x,y} and Y= {u,v}. Write down all the functions that can be defined from X to Y.How many of these are (i) one-one (ii) onto
and (iii)one-one and onto?
(c) Let X and Y be sets containing m and n elements respectively. (i) What is the total member of functions from X to Y (ii) How many
Functions from X to Y are one-one according as m,n, m.n, and m=n?
(d)Let R be the relation on the set R of real numbers such that a R biff a-b is an integer. Test Whether R is an equivalence relation. If so
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find the equivalence class of 1 and w.r.t. this equivalence relation.
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(e)Find the number of equivalence relations on X ={1,2.3}. [Hint: Each partition of a set givs an equivalence relation.]
(f)Find least positive integer x satisfying 276x + 128=4 (mod 7)
(g)Show that if R is an equivalence relation on X then domR= rngR=X.
Write the following relations in tabular from and determine their type:
(h)R= {x,y): 2x-y=0] on A={1,2,3,....,13}
(i)R= {x,y):x divides y} on A= {1,2,3,4,5,6}
(j)R= {(x,y): y ≤ x ≤ 4} on A= {1,2,3,4,5}
