Sets: Is A ⊆ B? A={x ∈ ℤ | x ≡ 7 (mod 8)} B={x ∈ ℤ | x ≡ 3 (mod 4)}

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In summary, we can see that A is a subset of B since every element in A can be written in the form of 4p + 3, which is also the form of elements in B. This shows that A ⊆ B, as desired.
  • #1
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A={x ∈ ℤ | x ≡ 7 (mod 8)}
B={x ∈ ℤ | x ≡ 3 (mod 4)}

Is A ⊆ B? Yes

Since x ∈ A, then xa = 7 + 8a = 8a + 7 = 2(4a + 3) +1. And since the ∈ B are of the form xb = 3 + 4b = 4b + 3 = 2(2b + 1) + 1, both ∈ A,B are odd. A ⊆ B since the ∈ of both sets are of 2p + 1. Q.E.D.

Is this correct?
 
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  • #2
No. You want to show that every element of A is an element of B. All you showed is that neither contains an even integer.

Try writing 7 + 8a in a form that shows it is in B.
 
  • #3
I don't know how to go about doing that.
 
  • #4
What is this character - ℤ ?

7 + 8a = 3 + 4 + 4(2a) = 3 + 4(1 + 2a)
 
  • #5
Integers
 
  • #6
So do you see how the conclusion follows? It is pretty straight forward now.
 
  • #7
Since x ∈ A, then x = 7 + 8a = 8a + 7 = 4(2a) + 4 + 3 = 4(2a + 1) + 3. And since the ∈ B are of the form x = 3 + 4b = 4b + 3, all x = 4p + 3 and A ⊆ B. Q.E.D.

Correct?
 

1. What is the difference between A and B?

The main difference between A and B is their definitions. A is a set of integers where each element is congruent to 7 mod 8, while B is a set of integers where each element is congruent to 3 mod 4.

2. How do you determine if A is a subset of B?

To determine if A is a subset of B, we need to check if all elements in A are also present in B. In this case, since both sets have the same definition, A is a subset of B.

3. What does the notation "x ∈ ℤ" mean?

The notation "x ∈ ℤ" means that x is an element of the set of integers. In other words, x is a whole number without any decimal values.

4. How do you read the notation "x ≡ 7 (mod 8)"?

The notation "x ≡ 7 (mod 8)" means that x is congruent to 7 mod 8. This means that when x is divided by 8, the remainder is 7.

5. Can you give an example of an element in A and B?

An example of an element in A is 15, because when 15 is divided by 8, the remainder is 7. An example of an element in B is 11, because when 11 is divided by 4, the remainder is 3.

Suggested for: Sets: Is A ⊆ B? A={x ∈ ℤ | x ≡ 7 (mod 8)} B={x ∈ ℤ | x ≡ 3 (mod 4)}

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