Explain why 1 + 1 equals 2 in the most complex way possible

By constructing arithmetics from Peano’s axioms (or equivalent).

Define  1  as  suc0 .

Define 2 as  suc1 .

Define addition as:  ∀∈ℕ,+0=  and  ∀,∈ℕ,+suc=(+) .

Prove that  suc=+1 . ( +1=+suc0=suc(+0)=suc ).

Therefore,  1+1=suc1=2 .

Then, prove that in any system which include a subset of inductive number which is compatible with Peano numbers, it is indeed compatible and the definitions of addition, 1, and 2 hold.

Second system:

Defining (natural) numbers as finite cardinals, and defining addition of two numbers  , :  + , as the cardinality of the union of two disjoint sets of cardinality    and    respectively.

We could define  1  as the cardinality of set  {{}} , and  2  as the cardinality of set  {{},{{}}} .

First I would prove that cardinality is an equivalence relationship.

Then I could prove that sets  {{}}  and  {{{}}}  are disjoint, each has cardinality 1 and the union has cardinality 2, which would fix my definitions.

Third system:

Use any other set of definitions and work from it. What should I define as 1? What should I define as 2? How I define addition?

For example, let’s have a field (a set with a commutative, associative, operation with identity property called addition, and a second commutativee, associative, operation with identity property that distributes the first one called multiplication) with total order which is closed by addition and multiplication. Let’s define  0  as the identity element of addition and  1  as the identity element of multiplication. Then find a way to define  2  differently than  1+1 , then prove that  2  is  1+1 . The tricky part is to use a coherent intuitive definition of  2  that is not  1+1

Step-by-step explanation:

so firstly, we need to isolate the y variables to be able to solve these inequalities. firstly, add 0.5x on both sides of the first inequality and subtract 1.5x on both sides of the second inequality:

now since the slope is positive for the first inequality, this means that the line going upwards belongs to the first inequality, and the line going downwards belongs to the second inequality.

next, since y ≥ in the first inequality, this means that the shaded region will be above the first inequality's line, thus shading regions a and b.

next, since y ≤ in the second inequality, this means that the shaded region is going to be below this line, thus shading regions a and c.

since both lines shade region a, this means that region a is the solution.

step-by-step explanation:

mark brainliest!

12

step-by-step explanation:

12 + 5 = 17

12x2=24

24-7=17