Adamson U
Discreet Math
Practice Quiz
Question:
1. Prove that the intersection of two countable sets is countable.
2. Prove that the complement of a countable set is uncountable.
3. Prove that the Cartesian product of two countable sets is countable.
4. Prove that the set of all real numbers is uncountable.
5. Prove that the set of all natural numbers is countable.
6. Prove that the set of all rational numbers is countable.
7. Prove that the set of all irrational numbers is uncountable.
8. Prove that the set of all algebraic numbers is countable.
9. Prove that the set of all transcendental numbers is uncountable.
10. Prove that the set of all polynomials with integer coefficients is countable.
Answer:
1. To prove that the union of two countable sets is countable, we can use the fact that a group
is countable if a one-to-one correspondence exists between its elements and the natural
numbers. Suppose we have two countable sets, A and B. In that case, we can create a one-to-
one correspondence between the details of the union of A and B and the natural numbers by
listing the elements of A first, followed by the features of B that are not in A. This creates a one-
to-one correspondence between the details of the union of A and B and the natural numbers,
so the union of A and B is countable.
A set is countable if there is a one-to-one correspondence between its elements and the
natural numbers. This means that each group component can be paired with a unique natural
number, and every natural number corresponds to a component of the set.
, Given two countable sets, A and B, we can create a one-to-one correspondence between the
elements of the union of A and B and the natural numbers by listing the features of A first,
followed by the details of B that are not in A. This is because the elements of A are already
countable and have one-to-one correspondence with the natural numbers. By adding the
elements of B that are not already in A, we are adding more features to our list, but we can still
pair each piece with a unique natural number.
By doing this, we have created a one-to-one correspondence between the elements of the
union of A and B and the natural numbers, which means that the association of A and B is also
countable.
2. To prove that the intersection of two countable sets is countable, we can use the fact that a
group is countable if a one-to-one correspondence exists between its elements and the natural
numbers. Suppose we have two countable sets, A and B. In that case, we can create a one-to-
one correspondence between the features of the intersection of A and B and the natural
numbers by listing the details of A that are also in B in any order. This creates a one-to-one
correspondence between the elements of the intersection of A and B and the natural numbers,
so the intersection of A and B is countable.
A set is countable if there is a one-to-one correspondence between its elements and the
natural numbers. This means that each group component can be paired with a unique natural
number, and every natural number corresponds to a component of the set.
Given two countable sets, A and B, we can create a one-to-one correspondence between the
elements of the intersection of A and B and the natural numbers by listing the details of A that
are also in B in any order. This is because both A and B elements are already countable and
have a one-to-one correspondence with the natural numbers within each set. By listing these
elements in any order, we can pair each piece with a unique natural number, which is the
definition of countable.
Discreet Math
Practice Quiz
Question:
1. Prove that the intersection of two countable sets is countable.
2. Prove that the complement of a countable set is uncountable.
3. Prove that the Cartesian product of two countable sets is countable.
4. Prove that the set of all real numbers is uncountable.
5. Prove that the set of all natural numbers is countable.
6. Prove that the set of all rational numbers is countable.
7. Prove that the set of all irrational numbers is uncountable.
8. Prove that the set of all algebraic numbers is countable.
9. Prove that the set of all transcendental numbers is uncountable.
10. Prove that the set of all polynomials with integer coefficients is countable.
Answer:
1. To prove that the union of two countable sets is countable, we can use the fact that a group
is countable if a one-to-one correspondence exists between its elements and the natural
numbers. Suppose we have two countable sets, A and B. In that case, we can create a one-to-
one correspondence between the details of the union of A and B and the natural numbers by
listing the elements of A first, followed by the features of B that are not in A. This creates a one-
to-one correspondence between the details of the union of A and B and the natural numbers,
so the union of A and B is countable.
A set is countable if there is a one-to-one correspondence between its elements and the
natural numbers. This means that each group component can be paired with a unique natural
number, and every natural number corresponds to a component of the set.
, Given two countable sets, A and B, we can create a one-to-one correspondence between the
elements of the union of A and B and the natural numbers by listing the features of A first,
followed by the details of B that are not in A. This is because the elements of A are already
countable and have one-to-one correspondence with the natural numbers. By adding the
elements of B that are not already in A, we are adding more features to our list, but we can still
pair each piece with a unique natural number.
By doing this, we have created a one-to-one correspondence between the elements of the
union of A and B and the natural numbers, which means that the association of A and B is also
countable.
2. To prove that the intersection of two countable sets is countable, we can use the fact that a
group is countable if a one-to-one correspondence exists between its elements and the natural
numbers. Suppose we have two countable sets, A and B. In that case, we can create a one-to-
one correspondence between the features of the intersection of A and B and the natural
numbers by listing the details of A that are also in B in any order. This creates a one-to-one
correspondence between the elements of the intersection of A and B and the natural numbers,
so the intersection of A and B is countable.
A set is countable if there is a one-to-one correspondence between its elements and the
natural numbers. This means that each group component can be paired with a unique natural
number, and every natural number corresponds to a component of the set.
Given two countable sets, A and B, we can create a one-to-one correspondence between the
elements of the intersection of A and B and the natural numbers by listing the details of A that
are also in B in any order. This is because both A and B elements are already countable and
have a one-to-one correspondence with the natural numbers within each set. By listing these
elements in any order, we can pair each piece with a unique natural number, which is the
definition of countable.
