The set {1, 2, 3, 5, 7, 8, 9} under multiplication modulo 10 is not a group. Given below are four possible reasons. Which one of them is false?a)It is not closedb)2 does not have an inversec)3 does not have an inversed)8 does not have an inverseCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Explanation:

To be a group, a set under an operation needs to satisfy the following four axioms:

1. Closure: For any two elements a and b in the set, their product (or sum) also belongs to the set.
2. Associativity: For any three elements a, b, and c in the set, (a*b)*c = a*(b*c).
3. Identity Element: There exists an element e in the set such that for any element a in the set, a*e = e*a = a.
4. Inverse Element: For any element a in the set, there exists an element b in the set such that a*b = b*a = e.

Let's check each of these axioms for the given set under multiplication modulo 10:

1. Closure: The product of any two elements in the set is also in the set, so this axiom is satisfied.
2. Associativity: Multiplication is associative, so this axiom is satisfied.
3. Identity Element: The element 1 is the identity element for multiplication, and it is in the set, so this axiom is satisfied.
4. Inverse Element: To find the inverse of an element a in the set, we need to find an element b such that a*b = b*a = 1 (mod 10).
- For 1, the inverse is 1 itself.
- For 2, there is no inverse because 2*5 = 10 = 0 (mod 10), not 1.
- For 3, there is no inverse because 3*7 = 21 = 1 (mod 10), not 1.
- For 5, the inverse is 5 itself.
- For 7, the inverse is 3.
- For 8, there is no inverse because 8*anything = even number, not 1.

So, we can see that the set does not satisfy the inverse element axiom for the element 3. Hence, option (c) '3 does not have an inverse' is false because 3 does have an inverse (7). The correct option should be (d) '8 does not have an inverse'.

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