Excercises

Excercise 1.3

1)If 81270 is a multiple of 3 or 7,that means that

81270=a*3 or 81270=b*7

That is,a=81270/3 or b=81270/7

Thus proved that 81270 is  either a multiple of 3 or 7

2)Given

                200000000=375*a

                Therefore 400000000=2*200000000=2*375*a

                Therefore it is proved that  if 20000000 is a multiple of 375,then 400000000 is a multiple of 375

3)200000000 is not a multiple of 375.

Irrelevant of the truth of the above statement ,the  statement that if 20000000 is a multiple of 375 then 40000000 is a multiple of 375 has to be proved...

4)Excercise 1.5.1

Counter example for 6n+1 is prime

Take n=4,thus 6n+1=6*4+1=25 ,which is not prime..

5)Exercie 1.5.2

If n is a multiple of 3 then n is a multiple of 6 true or false

Say a=3*n,where n is any natural number..

Therefore

B=6n,where n is any natural number,

ie,b=2*(3n)

and hence proved that if n is a multiple of 3 then n is a multiple of 6

6)Exercise2.2.1.a

A={2,4,8}

B={m|m is an even number}

A is thus a subset of    B

7)Ex 2.2.1.b

A={3,7,1025} B=(m|m=2ⁿ -1 for some number n}

A is not a subset of B as all the elements of A do not satisfy the condition in B

8)2.2.1.c

A={0},B=0

A is not a subset of B

9)A=0,B={0}

A is a subset of B ,as null set is the subset of any given set

10)2.2.2

A={a,b,c}

The subsets are

{a},{b},{c},{a,b},{b,c},{c,a},{a,b,c},0

Exercise 3.5

#Contrapositive statements

1a)If n is a multiple of 3 then n is not a multiple of 7

1b)If n is not a multiple of 4 then n is not a multiple of 12

2)Statement 1a is false and statement  1b is true

3)The converse of 1a and 1b are:

                If n is not a multiple of of 3, then n is a multiple of 7

                If n is a multiple of 4,then n is a multiple of 12

Thus the statement 1a is again false and 1b is true

4)If n+1 is prime, then  n  is not  a prime and n<3

Suppose 29 is not prime,this implies that there exists two numbers r and s such that r and s not equal to 1 and 29.

Contrapositive:If  r and s not equal 1 and 29,then 29 is not prime where 29=r*s

Inverse:If 29  is prime where 29=r*s,then  r and s is 1 and 29

Converse:If r and s is not  1 and 29,then 29 is not prime,where 29=r*s