Is the set \mathbb{z} a group under the operation a*b = a - b + ab ? justify your answer.

Step-by-step explanation:

Given that *is the binary operation in the sets of integers.

closure: a-b+ab is again an integer belongs to Z.  Hence closure is true.

Associativity:

The two are not equal.  Hence this cannot be a group as associtiavity does not hold good.

first option

a function is a relation in which an input is paired with exactly one output.

so, let's find the right answer.

in our first option, our inputs are january, february, and march. the outputs are 30, 28, and 25. each input has it's own output. this is correct!

our second option we know is incorrect, because ms. valverade has two outputs, dogs and cats. this is not a function.

our third option is incorrect as well. the yellow square has two ouputs, the square and the triangle.

lastly, in our fourth option, we see the number 7 two times with a different output. therefore, this is not a function.

remember: each input must be paired with only one output!