Chapter 1 Flashcards Preview

MAT 1362 > Chapter 1 > Flashcards

Flashcards in Chapter 1 Deck (16)
Loading flashcards...
1

commutativity of addition [axiom]

a + b = b + a

2

associativity of addition [axiom]

(a + b) + c = a + (b + c)

3

distributivity [axiom]

a * (b + c) = a * b + a * c

4

commutativity of multiplication [axiom]

a * b = b * a

5

associativity of multiplication [axiom]

(a * b) * c = a * (b * c)

6

additive identity [axiom]

There exists an interger 0 such that
a + 0 = a
for all a E Z.

The element 0 is called an additive identity.

7

multiplicative identity [axiom]

There exists an interger such that
1 =/= 0 and a * 1 = a
for all aEZ.

The element 1 is called a multiplicative identity.

8

additive inverse [axiom]

For each a E Z, there exists an interger, denoted -a, such that
a + (-a) = 0 .

The element -a is called the additive inverse of a.

9

cancellation property [axiom]

If a, b, c E Z,
a * b = a * c,
and a =/= 0,
then b = c .

10

reflexivity of (in)equality [axiom]

a = a
Does not apply to =/= .

11

symmetry of (in)equality [axiom]

If a = b, then b = a .
Also applies to =/= .

12

transitivity of (in)equality [axiom]

If a = b and b = c, then a = c .
Does not apply to =/= .

13

replacement property [axiom]

If a, b, c E Z and a = b,
then a + c = b + c .

14

uniqueness of the additive inverse [proposition]

If a, b E Z,
and a + b = 0 ,
then b = -a

15

uniqueness of the additive identity [proposition]

If a E Z has the property that
b + a = b for all b E Z,
then a = 0 .

16

uniqueness of the multiplicative identity [proposition]

If a E Z has the property that, for some nonzero b E Z,
b * a = b,
then a = 1 .