Algebraic Structures | MCQs

Explanation

The additive inverse of a number is what you add to a number to get zero.

In this context, 1 and -1 are additive inverses of each other, satisfying:

3 + (-1) = 2 and 1 - 3 = -2, showing the concept revolves around opposites in addition.

Explanation

Let's evaluate the expressions using the given definition of a * b:

3 * 4 = 3 + 4 + (3)(4) = 3 + 4 + 12 = 19

2 * 3 = 2 + 3 + (2)(3) = 2 + 3 + 6 = 11

Now, let's subtract:

3 * 4 - 2 * 3 = 19 - 11 = 8

Explanation

Given the operation a * b = a + b + 1:

Let's find the identity element 'e' such that a * e = a.

a * e = a + e + 1 = a

This implies e + 1 = 0, so e = -1.

Now, to find the inverse 'x' of 'a' such that a * x = e = -1:

a * x = a + x + 1 = -1

This implies a + x = -2.

x = -2 - a

x = -a - 2