MATH 109 Discussion 1

Necessary and Sufficient Conditions, Axiomatic Properties of the Real Numbers

2.3.1: + and - for real numbers

1. Commutativity:
2. Associativity:
3. Distributivity:
4. Zero Identity:
5. Unity:
6. Subtractivity: ( is defined as )
7. Division: (, is defined as )

In one word, 1-7 collectively say is a field.

Which of the following conditions are necessary for the positive integer $n$ to be divisible by 6 (proofs are not necessary)?

1. 3 divides
2. 9 divides
3. 12 divides
5. 6 divides
6. 2 divides and 3 divides
7. 2 divides or 3 divides

Answer

Necessary is the same as saying is divisible by 6 a condition 1, 5, 6, 7 Strong condition (6) implies weak condition (7)

Sufficient is the same as saying condition is divisible by 6

1. 3 divides - necessary
2. 9 divides - neither
3. 12 divides - sufficient
4. - sufficient
5. 6 divides - necessary, sufficient
6. 2 divides and 3 divides - necessary, sufficient
7. 2 divides or 3 divides - necessary

Use the properties of addition and multiplication of real numbers given in Properties 2.3.1 to deduce that, for all real numbers $a$ and $b$,

Answer

Problem 1.

Subtracting from both sides: Commutative:

Problem 2.

Subtracting : Commutative:

Problem 3.

and both solve the equation so