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prashantgolu
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if a set is closed and associative with respect to an operation * and both cancllation laws hold...prove that the set is a group wrt *.
Cancellation laws in group theory state that if two elements in a group have the same result after a group operation, then they are equal. In other words, if a * b = a * c, then b = c. This property allows for simplification of equations and easier manipulation of group elements.
Cancellation laws are often used in various fields of science and mathematics, such as physics, chemistry, and cryptography. For example, in physics, they can be used to simplify equations and solve for unknown variables. In cryptography, they are used to ensure the security of encrypted messages.
The left cancellation law states that if a * b = a * c, then b = c, while the right cancellation law states that if b * a = c * a, then b = c. Essentially, the left cancellation law applies when the element to the left of the group operation is the same, while the right cancellation law applies when the element to the right of the group operation is the same.
No, cancellation laws only apply in groups that are closed under the group operation. This means that the result of the operation must also be an element of the group. If the group is not closed, then cancellation laws may not hold.
Yes, cancellation laws can be applied to non-commutative groups, as long as the group is closed under the group operation. However, the order of the elements may affect the result, so the left and right cancellation laws may not be equivalent in this case.