MTH202 Assignment 2 solution

MTH202 Assignment 2 solution

solution of Q3 is on handouts page 147
..........
Q#01: Let X = {1, 5, 9}, Y = {3, 4, 7}
Define f : X Y by f(1) = 4, f(5) = 7, f(9) = 4
Marks = 2+2+1
Hint:
Chapter 16_Pg118.

(i) Is f one-one?
No, f is not one-to-one because all the elements of X are not mapped onto the Y elements. In X, 1 and 9 are mapped onto the same element 4 of Y.

(ii) Is f onto?
It’s also not a onto function because in Y, 3 is not img of any of X.


(iii) Does inverse of f exist?
No, f has not inverse because
A function f(x) has an inverse function if and only if f(x) is [COLOR=#0000FF !important]one to one[/COLOR] and it is not one to one. So, f has not inverse.



Q#02: Let A = {a, b, c} and R = {(a, c), (b, b), (c, a)} be a relation on A.

Determine whether R is reflexive, Symmetric, Transitive and anti-symmetric, or not.  Marks = 1+1+1+1

Solution:

Hint:
Chapter 12.
R is not reflexive.
Reason:
Because in R (a, a) and (c, c) are not exist. While in Reflexive Relation each element of A is must related to itself. So, in R is only (b, b) exist.
R is Symmetric.
Reason:
Because all the element of R are two ways. So, it’s symmetric.
R is not Transitive.
Reason:
According to [COLOR=#0000FF !important]definition[/COLOR] it should be must if a à b à c then a à c. so, it is not applying here.
R is not anti-symmetric.
Reason:
Because (c, a) & (a, c) ∈ R but c ≠ a

Q#03: Find the 36th term of the arithmetic [COLOR=#0000FF !important]sequence[/COLOR] whose 3rd term is 7 and 8th term is 17. Marks 6
Solution:

If
First term = a
Common difference = d
3rd term num = n = 3
8th term num = n = 8

So,
7 = a + (3-1) d
7 = a +2d------- (1)
17 = a + (8-1) d
17 = a +7d------- (2)
And now we find
d = ?
a =?
For finding the value of ‘d’ we subtract [COLOR=#0000FF !important]equation[/COLOR] 1st from 2nd

17 = a + 7d
7 = a + 2d
-----------------
10 = 5d
10/2 = d
d = 2
For finding the value of ‘a’, we put the value of ‘d’ in equation (1).

7 = a + 2d
7 = a + 2(2)
7 = a + 4
7- 4 = a
a = 3
We know that
So, the value of 36th term is
= 3 + (36-1) 2
= 3 + (35) 2
= 3+70
.............................

Q#01: Let X = {1, 5, 9}, Y = {3, 4, 7}
Define f : X Y by f(1) = 4, f(5) = 7, f(9) = 4
Marks = 2+2+1
Hint:
Chapter 16_Pg118.

(i) Is f one-one?
No, f is not one-to-one because all the elements of X are not mapped onto the Y elements. In X, 1 and 9 are mapped onto the same element 4 of Y.

(ii) Is f onto?
It’s also not a onto function because in Y, 3 is not img of any element of X.


(iii) Does inverse of f exist?
No, f has not inverse because
A function f(x) has an inverse function if and only if f(x) is one to one and it is not one to one. So, f has not inverse.

...................
Q#02: Let A = {a, b, c} and R = {(a, c), (b, b), (c, a)} be a relation on A.
Determine whether R is reflexive, Symmetric, Transitive and anti-symmetric, or not.
Marks = 1+1+1+1
Solution:

Hint:
Chapter 12.
R is not reflexive.
Reason:
Because in R (a, a) and (c, c) are not exist. While in Reflexive Relation each element of A is must related to itself. So, in R is only (b, b) exist.
R is Symmetric.
Reason:
Because all the element of R are two ways. So, it’s symmetric.
R is not Transitive.
Reason:
According to definition it should be must if a à b à c then a à c. so, it is not applying here.
R is not anti-symmetric.
Reason:
Because (c, a) & (a, c) ∈ R but c ≠ a
.................
Q#03: Find the 36th term of the arithmetic sequence whose 3rd term is 7 and 8th term is 17. Marks 6
Solution:
If
First term = a
Common difference = d
3rd term num = n = 3
8th term num = n = 8

So,
7 = a + (3-1) d
7 = a +2d------- (1)
17 = a + (8-1) d
17 = a +7d------- (2)
And now we find
d = ?
a =?
For finding the value of ‘d’ we subtract equation 1st from 2nd

17 = a + 7d
7 = a + 2d
-----------------
10 = 5d
10/2 = d
d = 2
For finding the value of ‘a’, we put the value of ‘d’ in equation (1).

7 = a + 2d
7 = a + 2(2)
7 = a + 4
7- 4 = a
a = 3
We know that
So, the value of 36th term is
= 3 + (36-1) 2
= 3 + (35) 2
= 3+70