Properties and Identities (Equality) Samantha N. Ame
COMMUTATIVE PROPERTY
Order of factors does not affect the
sum/ product.
a + b = b + a
2 + 3 = 3 + 2
ASSOCIATIVE PROPERTY
Grouping of factors does not affect the
sum/ product.
(a + b) + c = a + (b + c)
(2 + 3) + 4 = 2 + (3 + 4)
(a × b) × c = a × (b × c)
(2 × 3) × 4 = 2 × (3 × 4)
DISTRIBUTIVE PROPERTY
Multiplication operation can be
distributed over an add’n/sub’n op.
a(b + c) = ab + ac
2(3 + 4) = (2 x 3) + (2 x 4)
ADDITIVE IDENTITY
Adding zero to any number leaves it
unchanged.
a + 0 = a
2 + 0 = 2
MULTIPLICATIVE IDENTITY
Multiplying 1 to any number leaves it
unchanged.
a x 1 = a
2 x 1 = 2
INVERSE PROPERTY
Additive inverse of a is –a
inverse of 2 is -2
Multiplicative inverse of a is 1/a
inverse of 2 is 1/2
REFLEXIVE PROPERTY
Any number is equal to itself
a = a 2 = 2
SYMMETRIC PROPERTY
If a=b, then b=a
if 2 + 3 = 5, then 5 = 2 + 3
TRANSITIVE PROPERTY
if a = b and b = c, then a = c
given:
a= 2+3
b=5
c=7-2
if a=b : 2+3=5
& b=c: 5= 7-2
then a=c: 2+3= 7-2
SUBSTITUTION PROPERTY
If a=b, then b can be substituted for a in
any expression.
x + y = 10 where x= 4
using substitution property,
4 + y= 10
4 + y - 4 = 10 – 4 (APE/SPE)
y=6
CLOSURE PROPERTY
the set of integers is closed under
addition, so if a and b are integers, then
a + b is also an integer.
useful tool for determining whether the operation
is valid and can be used to reason abt.
properties even if we don’t know the final answer
yet.
consider the set of whole numbers,
which includes all positive integers
(1, 2, 3, ...) and zero (0).
If we take any two whole numbers,
say 3 and 5, and perform the
operation of addition, we get:
3 + 5 = 8
Since 8 is also a whole number, the
closure property is satisfied.
