(a) Yes, the set {1,2,3,4,5,7} is a group under addition modulo 7. Here's why:

1. Closure: If you add any two numbers in the set, and then take that sum modulo 7, you will get another number that is still in the set. For example, 5 + 7 = 12, and 12 modulo 7 is 5, which is in the set.

2. Associativity: The operation of addition is associative. This means that for any a, b, and c in the set, (a + b) + c is equal to a + (b + c).

3. Identity: The number 0 is the identity element for addition. However, 0 is not in the set, so the set does not contain the identity element under addition modulo 7.

4. Inverse: For every element a in the set, there is another element b in the set such that a + b = 0 (mod 7). However, since 0 is not in the set, the set does not contain inverses for all elements.

So, the set {1,2,3,4,5,7} is not a group under addition modulo 7 because it does not contain the identity element and inverses for all elements.

(b) Yes, the set {1,2,3,4,5,7} is a group under multiplication modulo 7. Here's why:

1. Closure: If you multiply any two numbers in the set, and then take that product modulo 7, you will get another number that is still in the set. For example, 5 * 7 = 35, and 35 modulo 7 is 0, which is not in the set.

2. Associativity: The operation of multiplication is associative. This means that for any a, b, and c in the set, (a * b) * c is equal to a * (b * c).

3. Identity: The number 1 is the identity element for multiplication. This means that for any a in the set, 1 * a = a * 1 = a. So, the set does contain the identity element under multiplication modulo 7.

4. Inverse: For every element a in the set, there is another element b in the set such that a * b = 1 (mod 7). However, the number 7 does not have an inverse in the set, so the set does not contain inverses for all elements.

So, the set {1,2,3,4,5,7} is not a group under multiplication modulo 7 because it does not contain inverses for all elements.

Question

(a) Yes, the set {1,2,3,4,5,7} is a group under addition modulo 7. Here's why:

1. Closure: If you add any two numbers in the set, and then take that sum modulo 7, you will get another number that is still in the set. For example, 5 + 7 = 12, and 12 modulo 7 is 5, which is in the set.

2. Associativity: The operation of addition is associative. This means that for any a, b, and c in the set, (a + b) + c is equal to a + (b + c).

3. Identity: The number 0 is the identity element for addition. However, 0 is not in the set, so the set does not contain the identity element under addition modulo 7.

4. Inverse: For every element a in the set, there is another element b in the set such that a + b = 0 (mod 7). However, since 0 is not in the set, the set does not contain inverses for all elements.

So, the set {1,2,3,4,5,7} is not a group under addition modulo 7 because it does not contain the identity element and inverses for all elements.

(b) Yes, the set {1,2,3,4,5,7} is a group under multiplication modulo 7. Here's why:

1. Closure: If you multiply any two numbers in the set, and then take that product modulo 7, you will get another number that is still in the set. For example, 5 * 7 = 35, and 35 modulo 7 is 0, which is not in the set.

2. Associativity: The operation of multiplication is associative. This means that for any a, b, and c in the set, (a * b) * c is equal to a * (b * c).

3. Identity: The number 1 is the identity element for multiplication. This means that for any a in the set, 1 * a = a * 1 = a. So, the set does contain the identity element under multiplication modulo 7.

4. Inverse: For every element a in the set, there is another element b in the set such that a * b = 1 (mod 7). However, the number 7 does not have an inverse in the set, so the set does not contain inverses for all elements.

So, the set {1,2,3,4,5,7} is not a group under multiplication modulo 7 because it does not contain inverses for all elements.

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