Set theory, intersection of two sets

In summary, the problem is to find the numbers that are both prime and congruent to 1 modulo 8. This requires creating a long list of numbers and checking for matches, as no other known method exists.

Homework Statement

We have the set D which consists of x, where x is a prime number. We also have the set F, which consists of x, belongs to the natural numbers (positive numbers 1, 2, 3, 4, 5..) that is congruent with 1 (modulo 8). What numbers are in the intersection of these two sets?

The Attempt at a Solution

So the set F consists of numbers that when divided by 8 gives the remainder 1. So, 1, 9, 17, 25, 33, 41, and so on. The set D is prime numbers. So: 2 3 5 7 11 13 17 19 23 29 31 37 and so on. I am to find the numbers that are in both of these sets. But how can I do this? I have no idea except for making a long list of numbers and matching them.

Homework Statement

We have the set D which consists of x, where x is a prime number. We also have the set F, which consists of x, belongs to the natural numbers (positive numbers 1, 2, 3, 4, 5..) that is congruent with 1 (modulo 8). What numbers are in the intersection of these two sets?

The Attempt at a Solution

So the set F consists of numbers that when divided by 8 gives the remainder 1. So, 1, 9, 17, 25, 33, 41, and so on. The set D is prime numbers. So: 2 3 5 7 11 13 17 19 23 29 31 37 and so on. I am to find the numbers that are in both of these sets. But how can I do this? I have no idea except for making a long list of numbers and matching them.

It would have to be a very very long list. The number of primes of the form 8k+1 is known to be infinite. I don't know any nice form to express the intersection besides that.

1. What is a set in set theory?

A set in set theory is a collection of distinct objects, called elements, that are grouped together based on a common characteristic or property. Sets are denoted by curly brackets { } and each element is separated by a comma. For example, the set of all even numbers can be written as {2, 4, 6, 8, ...}.

2. How do you represent a set in set theory?

In set theory, a set can be represented in three different ways: verbally, using the roster or list method, and using set-builder notation. Verbal representation describes the elements of a set using words, while the roster method lists all the elements of a finite set within curly brackets. Set-builder notation uses a variable, a condition, and a colon to describe the elements of a set. For example, the set of all positive even numbers can be represented as {x | x is a positive even number}.

3. What is the intersection of two sets?

The intersection of two sets is the set of all elements that are common to both sets. It is denoted by the symbol ∩ and is read as "intersection". For example, if set A = {1, 2, 3} and set B = {2, 3, 4}, then the intersection of A and B is {2, 3}.

4. How do you find the intersection of two sets?

To find the intersection of two sets, you can use the Venn diagram method or the set-builder notation method. In the Venn diagram method, draw two overlapping circles to represent the two sets and the intersection will be the region where the circles overlap. In set-builder notation, write out the elements of both sets and the common elements will be the ones included in both sets.

5. What are the properties of the intersection of two sets?

The intersection of two sets has the following properties:

• The intersection is commutative, meaning that A ∩ B = B ∩ A.
• The intersection is associative, meaning that (A ∩ B) ∩ C = A ∩ (B ∩ C).
• The intersection is distributive, meaning that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
• The intersection of a set with the empty set is always equal to the empty set, meaning that A ∩ ∅ = ∅.

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