Math 492/592
Homework 1.4 Identities, Inverses and Closure
1. Identity Table
Write the identity element (e.g. 0 or 1) and corresponding symbolic statement
when the set has an identity element and Identity Property for that operation.
Write none and give a counterexample or short explanation when there is no
identity element and/or property for that set and that operation.
Set
Addition
Subtraction
Multiplication
Division
W
0
0 + a = a + 0 =a
∀a ∈ W
None
0–3≠3
1
1×a=a×1=a
∀a ∈ W
None
5 ÷ 1 ≠ 1÷ 5
Z
Q
R
W*
Z*
Q*
R*
Topic 1.4 HW, page 1
2. Inverses Table
Give the general “inverse element” and corresponding symbolic statement
when the set has an Inverse Property for that operation. Write none and give a
counterexample or brief explanation when there is no inverse property for that
set and that operation. W is done as an example.
Set
Addition
Subtraction
Multiplication
Division
W
None
No -a ∈ W
a
a - a = a - a =0
∀a ∈ W
None, 0 has no
inverse in W
None
5 ÷ 1 = 5 but
1 ÷ 5 =1/5 ≠ 5
Z
-a
-a + a = a + -a =0
∀a ∈ Z
Q
R
W*
Z*
Q*
R*
Topic 1.4 HW, page 2
1/a
1/a×a = a×1/a =1
∀a ∈ z
3. Closure Table
Write “closed” and the corresponding symbolic statement when the set is
closed with respect to that operation. Write “not closed” and give a
counterexample or brief explanation when the set is not closed that operation.
W is done as an example.
Set
Addition
Subtraction
Multiplication
Division
W
Closed
a+b∈W
∀a, b ∈ W
Not closed
3 – 5 = -2  W
Closed
a×b∈W
∀∈W
Not closed
1 ÷ 5 =1/5  W
Z
Q
R
W*
Z*
Q*
R*
Topic 1.4 HW, page 3
5. (Adapted from a 9th grade textbook.1) Assume Z is closed under multiplication
and addition.
a. Prove the set of rational numbers Q is closed under addition by completing
the following. Fill in the five boxes and the blank.
Step
Let a and b be any rational numbers.
We want to show a + b is in Q.
a = p/q and b = r/s where p, q, r, s are in Z with q
and s nonzero
p r
ab 
q s

ps r

qs s

qs

qs
Justification
Begin the proof.
State the goal.
By one of the definitions of
rational numbers
Substitution
Find a common denominator

Add the fractions
qs
The numerator and denominator of a + b are both integers and the denominator is
not 0 because ____________________________________________________.
Therefore, a + b is in Q.
b. Write a similar proof to show Q is closed under multiplication.
1
On Core Mathematics: Grade 9, Houghton-Mifflin Harcourt 2013, Unit 1 Lesson 1
Topic 1.4 HW, page 4
6. Define the following new (weird but cool) operation # on Q.
∀ a, b  Q, a # b = a + b + 4
a. Calculate (give answers as whole numbers or ratios of integers, but not as
decimals). Follow order of operations and do the calculation inside the ( )s
first. Show your work.
i) 2 # 3
ii) 3 # 2
1
3
iii) # 
2
7
3
1
iv)  #   # 4
7
2
v)
1  3

#   # 4
2  7

For the following sets of symbolic statements and proofs (parts b – f):
 Don’t write in other operations (except for the +4)
 In every case, use a and b as your general rational numbers (that is start with
“let a, b  Q”)
 You can use properties of Q under + to justify proof steps
b. Write the general symbolic statement for what it means to say Q has the
Commutative Property for #.
Does Q have the Commutative Property for #? Prove it symbolically or give
a counterexample.
c. Write the general symbolic statement for what it means to say Q has the
Associative Property for #.
Does Q have the Associative Property for #? Prove it symbolically or give a
counterexample.
d. Write the general symbolic statement for what it means to say Q has the
Identity Property for #.
Does Q have the Identity Property for #? Prove it symbolically or give a
counterexample. Of course, you’ll have to give the Identity element for Q
under # in this proof.
e. Write the general symbolic statement for what it means to say Q has the
Inverses Property for #.
Does Q have the Inverses Property for the operation #? Prove it symbolically
or give a counterexample. Of course, you’ll have to give the general form for
the inverse element for a  Q under # in this proof.)
Topic 1.4 HW, page 5
f. Write the symbolic statement for what it means to say Q has the Closure
Property for #.
Does Q have the Closure Property for #? Prove it symbolically or give a
counterexample.
7. Extend (reduce?) the following operation # to Q*.
∀ a, b  Q*, a # b = a + b + 4
a. Is # commutative in Q*? Explain why or give a counterexample.
b. Does Q* have the Identity property for #? Explain.
c. Is Q* closed under #? Explain why or give a counterexample.
Topic 1.4 HW, page 6