A = { x ∈ Z : 0 ≤ x ≤ 12} = {0,1,2,3,4,5,6,7,8,9,10,11,12}
(i) R = { (a,b) : |a - b| is a multiple of 4}
For any element a ∈A, we have (a, a) ∈ R as |a - a = 0|is a multiple of 4.
∴R is reflexive.
Now, let (a, b) ∈ R ⇒ |a - b| is a multiple of 4.
⇒ |-(a - b)| = ⇒ |b - a| is a multiple of 4.
⇒ (b, a) ∈ R
∴R is symmetric.
Now, let (a, b), (b, c) ∈ R.
⇒ |(a - b)| is a multiple of 4 and |(b - c)| is a multiple of 4.
⇒ (a - b) is a multiple of 4 and (b - c) is a multiple of 4.
⇒ (a - c) = (a – b) + (b – c) is a multiple of 4.
⇒ |a - c| is a multiple of 4.
⇒ (a, c) ∈R
∴ R is transitive.
Hence, R is an equivalence relation.
The set of elements related to 1 is {1, 5, 9} since
|1 - 1| = 0 is a multiple of 4,
|5 - 1| = 4 is a multiple of 4, and
|9 - 1| = 8 is a multiple of 4.
(ii) R = {(a, b): a = b}
For any element a ∈A, we have (a, a) ∈ R, since a = a.
∴R is reflexive.
Now, let (a, b) ∈ R.
⇒ a = b
⇒ b = a
⇒ (b, a) ∈ R
∴R is symmetric.
Now, let (a, b) ∈ R and (b, c) ∈ R.
⇒ a = b and b = c
⇒ a = c
⇒ (a, c) ∈ R
∴ R is transitive.
Hence, R is an equivalence relation.
The elements in R that are related to 1 will be those elements from set A which are equal to 1.
Hence, the set of elements related to 1 is {1}.
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
Let f : R → R be defined as f(x) = 3x. Choose the correct answer.
(A) f is one-one onto
(B) f is many-one onto
(C) f is one-one but not onto
(D) f is neither one-one nor onto.
A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.
Answer the following as true or false.
\begin{align}(i) \overrightarrow{a}\; and\; \overrightarrow{-a}\; are\; collinear.\end{align}
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
Find gof and fog, if
(i) f(x) = | x | and g(x) = | 5x – 2 |
(ii) f(x) = 8x3 and g(x) = x1/3 .
Let f : R → R be defined as f(x) = 3x. Choose the correct answer.
(A) f is one-one onto
(B) f is many-one onto
(C) f is one-one but not onto
(D) f is neither one-one nor onto.
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