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) = x4. 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.
y = x2 + 2x + C : y' - 2x - 2 = 0
Show that the function f : R* → R* defined by f(x) = 1/x is one-one and onto,where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R* ?
Maximise Z = 3x + 4y
Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0
Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.
y = cosx + C : y' + sinx = 0
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