“Discrete Structures”
“Q2: Determine whether each of these functions from {a,b,c,d} to itself is onto and/or one-to-one. a) f(a)=b, f(b)=a, f(c)=c, f(d)=d
a

b

c

d

a

b

c

d

a

b

c

d
a

b

c

d

The above Function is both one-to-one and onto, therefore Its called a BIJECTION Function

b) f(a)=b, f(b)=b, f(c)=d, f(d)=c
a

b

c

d

a

b

c

d

a

b

c

d
a

b

c

d

The above Function is

c) a

b

c

d
a

b

c

d
a

b

c

d
a

b

c

d
f(a)=d, f(b)=b, f(c)=c, f(d)=d

The above Function is Neither On to nor one-to-one
“Q6: Take two sets A and B of your choice and determine the following: A = {0,1,2,3,4}
B = {2,4,6,8,10}
a) A∪B
= {0,1,2,3,4,6,8,10}

b) A∩B
= {2,4}

c) A-B
= {0,1,3}

d) B-A
= {6,8,10}

“Q4: Let S = {-1,0,2,4,7}. Find f(S) if the following conditions are followed” a)fx=1
As there are no values to substitute x with therefore this function shall remain the same regardless of the number of values

b)fx=2x+1
One by one the above mentioned set values are put into the function =2-1+1 =22+1 =5
= 1 =24+1 =9
=20+1
=1 S = {1, 1, 5, 9, 15}

=27+1
=15

c) fx=|x5|
Putting the set values into the above equation
|-15| = 15

05 =0

25 =0.4 S= {0.5, 0, 0.4, 0.8, 1.4}

45=0.8

75=1.4

e) fx=|x2+13|
Putting the values of the Set S in the above equations

-12+13=23

02+13=13
S = {2/3, 1/3, 5/3, 17/3, 50/3} 22+13=53

42+13=173

72+13=503

“Q8: Construct a proof that
a) If m is odd, then m^2 is odd
b) for all integers m and n, if m is even and n is odd, the m+n is odd” a) If m is odd, then m=2k + 1
We have to prove that m^2 is odd, so squaring the above mentioned equation we get =(2k + 1)^2
= 4k2+1+2(2k)(1)
=4k2+1+4k
Put in any odd value in the above equation
= 452+1+4(5)
=425+1+20
=121

Hence proved that the square of the value of m which is odd is also odd

b) Since m is odd (2k+1)
And n is even (2k)
Also since m+n is to be proved odd
=m+n
=2k + 1 + 2k
=4k + 1
Put any integer in the above function
4(5) + 1 4(4) + 1
=21 =21

4(10) + 1 4(7) + 1
=41 =29

“Q7: For each of these sets of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises.”

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