Prove one the two following theorems of Cantor: The set of rational numbers is a

Question

Prove one the two following theorems of Cantor: The set of rational numbers is a countable set; The set of real numbers is an uncountable set. Prove one of the two Laws of De Morgan from set theory: Give the Masse diagram for the divisors of each number: 51; 256. Use the Euclidean algorithm to find the greatest common divisor of the pair of numbers and write the greatest common divisor as an integral linear combination of the integers. (175.343); (756, 210). Express the logic statement in disjunctive normal form: Solve the system of linear congruences: {x equivalence 1 mod 4 x equivalence mod 9}

Explanation / Answer

2nd question:

a. (A B)' = A' U B'

Let M = (A B)' and N = A' U B'

Let x be an arbitrary element of M then x M x (A B)'

x (A B)

x A or x B

x A' or x B'

x A' U B'

x N

Therefore, M N …………….. (i)

Again, let y be an arbitrary element of N then y N y A' U B'

y A' or y B'

y A or y B

y (A B)

y (A B)'

y M

Therefore, N M …………….. (ii)

Now combine (i) and (ii) we get; M = N i.e. (A B)' = A' U B'

b) (A U B)' = A' B'

Let P = (A U B)' and Q = A' B'

Let x be an arbitrary element of P then x P x (A U B)'

x (A U B)

x A and x B

x A' and x B'

x A' B'

x Q

Therefore, P Q …………….. (i)

Again, let y be an arbitrary element of Q then y Q y A' B'

y A' and y B'

y A and y B

y (A U B)

y (A U B)'

y P

Therefore, Q P …………….. (ii)

Now combine (i) and (ii) we get; P = Q i.e. (A U B)' = A' B'

Question 6:

x= 1 mod 4

so lets say x=1+4t

x=8 mod 9 ===> 1+4t=8 mod 9

===> 4t= 7 mod 9

====> 4t= (9+7) mod 9

=====> 4t=16 mod 9

=====> t= 4 mod 9

======> t= 4+9s

since x=1+4t

====> x=1+4(4+9s)

====> x=1+16+36s

====> x=17+36 s where s is an integer

so solution is x= 17+36 s

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