Section 1.3: Subsets
We will start the section learning the definition
of two terms subset and proper subset. For
some reason this concept is missed a lot on
tests. Pay close attention to the difference in
the definitions.
Definition of Subset ( ⊆ )
Set A is a subset of set B, symbolized A ⊆ B, if
and only if all the elements of set A are also
elements of set B. Using the ⊆ symbol in a
problem allows the two sets to be identical
(equal)
Note that A is a subset of set B if the following
two conditions hold:
1. A is first and foremost a SET. (A can’t be
a subset if it isn’t a set.)
2. If x ∈ A, then x ∈ B. (this means every
element inside A is also an element inside
B)
Example: Is A ⊆ B given these sets: A =
{a,b,c,d} and B = {a,b}?
Let’s check the rules.
First A is a set because it is in set braces, so
rule one is satisfied.
Second is every element in set A also in set
B, no since A has elements “c” and “d” that
B doesn’t have so rule two is NOT satisfied.
Both rules are NOT satisfied. A is not a
subset of B
Answer: No
Example: Is S ⊆ T: where S = {1,2,3} and T =
{1,2,3,4}?
I changed the letters up, but the rules will still
work.
Let’s check the rules.
First S is a set because it is in set braces, so
rule one is satisfied.
Second is every element in set S also in set T,
yes so rule two is satisfied.
Both rules are satisfied so S ⊆ T
Answer: Yes, S ⊆ T
Example: Is T ⊆ S given these sets: S =
{a,b,c,d} and T = {a,b}?
Let’s check the rules.
First T is a set because it is in set braces, so
rule one is satisfied.
Second is every element in set T also in set
S, yes so rule two is satisfied.
Both rules are satisfied so T ⊆ S
Answer: Yes, T ⊆ S
Example: Is C ⊆ D given the sets: C = {1,2,3}
and D = {3,1,2}?
First C is a set, so rule one is satisfied.
Second every element of C is also an element
of D so rule 2 is satisfied.
Answer: Yes, C ⊆ D
It wasn’t asked but it is also true that D ⊆ C.
Sets can be subsets of each other if the sets
are equal.
Example: Is A ⊆ B given A = 1,2 and B =
{1,2,3}?
First A is NOT a set as it is not in set braces.
Rule 1 is not satisfied.
I can stop checking rules once one of the
rules is violated and simply answer no.
Answer: NO
Definition of Proper Subset ( ⊂ )
Set A is a proper subset of set B, symbolized by
A ⊂ B, if and only if the following three
conditions hold:
1. A is a set.
2. Every element of A is also an element of B.
3. A ≠ B.
Note: Basically this symbol is true when A is a
subset of B, but not equal to B.
Example: Is A ⊂ B: Where A = {1,2,3} and B =
{1,2,3,4}?
Let’s check the rules.
First A is a set because it is in set braces, so
rule one is satisfied.
Second is every element in set A also in set
B, yes so rule two is satisfied.
Third are the sets equal, they are not so rule
three is satisfied.
All rules are satisfied so A ⊂ B
Answer: Yes, A ⊂ B (thus set A is a proper
subset of B)
Example: Is S⊂ T given these sets: S = {a,b,c,d}
and T = {a,b}?
Let’s check the rules.
First S is a set because it is in set braces, so
rule one is satisfied.
Second is every element in set S also in set
T, no since S has elements c and d that T
doesn’t have so rule two is NOT satisfied.
I don’t need to check rule three as rule 2
has been violated.
All rules are NOT satisfied so S is not a
proper subset of T
Answer: No
Example: Is T ⊂ S given these sets: S =
{a,b,c,d} and T = {a,b}?
Let’s check the rules.
First T is a set because it is in set braces, so
rule one is satisfied.
Second is every element in set T also in set
S, yes so rule two is satisfied.
Third S does not equal T so rule 3 is
satisfied.
All rules are satisfied so T ⊂ S (that is T is a
proper subset of S)
Answer: Yes, T ⊂ S
Example: Is C ⊂ D given the sets: C = {1,2,3}
and D = {3,1,2}
First C is a set, so rule one is satisfied.
Second, every element of C is also an element
of D so rule 2 is satisfied.
Third the sets are equal, so rule three is NOT
satisfied.
Answer: No
Example: Is A ⊂ B given A = 1,2 and B = {1,2,3}
First A is NOT a set as it is not in set braces.
Rule 1 is not satisfied.
I don’t need to check rules 2 and 3 once rule 1
is not satisfied.
Answer: NO
We didn’t do much with the empty set in
section 1.1. We will do a bit with it now.
Remember that the empty set is a set with no
elements. It can be expressed in two ways.
Both ∅ and { } are representations of an empty
set.
Example: Fact: The empty set is a subset of
every set.
This will show { } ⊂ A for every set A. Also { }
⊆ A for every set A:
Rule 1 is satisfied as the empty set is a set.
Rule 2 is satisfied as everything in the empty set
is in every set A. For this rule to not be satisfied
there has to be something in the empty set that
is not in set A. Since there is nothing in the left
set this rule is satisfied.
Rule 3 is satisfied as long as A has something in
it the sets will not be equal.
So { } ⊂ A and { } ⊆ A are true for every set A
This tells me that the empty set is both a subset
and a proper subset of every set.
Example: Is C ⊂ D given the sets: C = { } and D
= {3, 1, 2}
Answer: yes, C ⊂ D (since the empty set is a
proper subset of every set)
Homework #1-18: Answer true or false, if false
give the reason
1) Is A ⊆ B given A = silver, B = {gold, silver,
diamond}
2) Is A⊆ B given A = Phoenix, B = {Phoenix,
Glendale, Peoria, Scottsdale}
3) Is C ⊆ D given C = {Phoenix}, D = {Phoenix,
Glendale, Peoria, Scottsdale}
4) Is C ⊆ D given C = {silver}, D = { gold, silver,
diamond}
5) Is A ⊆ B given A = {2,3} , B = {1,2,3,4,5}
6) Is A ⊆ B given A = 2,3 B = {1,2,3,4,5}
7) Is A ⊆ B given A = a, B = {𝑥|𝑥 𝑖𝑠 𝑎 𝑣𝑜𝑤𝑒𝑙}
8) Is A⊆ B given A= {a}, B = {𝑥|𝑥 𝑖𝑠 𝑎 𝑣𝑜𝑤𝑒𝑙 }
9) Is A ⊆ B given A = { }, B = {1,2,3,4,5}
10) Is S ⊆ T given S = ∅, T ={1,2,3}
11) Is S ⊂ T given S= ∅, T = {1,2,3,4,5}
12) Is S ⊂ T given S = { }, T = {1,2}
13) Is A ⊂ B given A = {1,2,3}, B = {3,2,1}
14) Is A ⊂ B given A = {a,b,c,d}, B = {d,c,b,a}
15) Is C ⊂ D given C = {1,2,3,4,5}, D = {1,2,3,4}
16) Is C ⊂ D given C = {w,x,y}, D = {x,y,z}
17) Is A ⊂ B given A = { 4,3,2,1 }, B = {1,2,3,4,5}
18) Is A ⊂ B given A = { e,b,c,d,a },
B = {a,b,c,d,e,f}
Example: Given set A = {1,2,3} and B = {3,2,1}
Determine which of these are true. (Choose
every answer that is true, in many instances
there will be more than one correct choice.)
A = B, A ⊆ B,
of these
B⊆A, A ⊂ B, B⊂A, or none
I will examine each symbol one at a time.
A = B: This is true, to be equal you need to
have exactly the same elements.
A ⊆ B: This is true as the two rules are
satisfied.
First A is a set because it is in set braces, so rule
one is satisfied.
Second is every element in set A also in set
B, yes so rule two is satisfied.
B ⊆ A: This is true as the two rules are
satisfied.
First B is a set because it is in set braces, so
rule one is satisfied.
Second is every element in set B also in set A,
yes so rule two is satisfied.
A ⊂ B: This is not true as this rule doesn’t allow
the sets to be equal.
B ⊂ A: This is not true as this rule doesn’t allow
the sets to be equal.
Answer: A = B,
A ⊆ B,
B⊆A
Example: Given set A = {1,2} and B = {3,2,1}
Determine which of these are true. (Choose
every answer that is true, in many instances
there will be more than one correct choice.)
A = B, A ⊆ B,
of these
B⊆A, A ⊂ B, B⊂A, or none
I will examine each symbol one at a time.
A = B: This is NOT true, to be equal you need to
have exactly the same elements.
A ⊆ B: This is true as the two rules are
satisfied.
First A is a set because it is in set braces, so
rule one is satisfied.
Second is every element in set A also in set B,
yes so rule two is satisfied.
B ⊆ A: This is NOT true as rule 2 is not satisfied.
First B is a set because it is in set braces, so
rule one is satisfied.
Second B has a 3 in it that A doesn’t, so rule 2
is not satisfied
A ⊂ B: This is true as all three rules are
satisfied.
First A is a set because it is in set braces, so rule
one is satisfied.
Second is every element in set A also in set B,
yes so rule two is satisfied.
Third A and B are not equal so rule three is
satisfied.
B ⊂ A: This is NOT true as rule 2 is not satisfied.
First B is a set because it is in set braces, so
rule one is satisfied.
Second B has a 3 in it that A doesn’t, so rule 2 is
not satisfied.
Answer: A ⊆ B, A ⊂ B
Homework #19 – 34: Determine which of these
are true. (Choose every answer that is true, in
many instances there will be more than one
correct choice.)
A = B, A ⊆ B,
none of these
B ⊆ A, A ⊂ B, B ⊂ A, or
19) A = {Trix, Captain Crunch, Rice Krispees}
B = {Rice Krispees}
20) A = {red, white, blue}
B = {red, blue}
Homework #19 – 34: Determine which of these
are true. (Choose every answer that is true, in
many instances there will be more than one
correct choice.)
A = B, A ⊆ B,
none of these
21) A = {5,7,9}
B ⊆ A, A ⊂ B, B ⊂ A, or
B = {9, 5, 7}
22) A = {a,b,c} B = {c, b, a}
Homework #19 – 34: Determine which of these
are true. (Choose every answer that is true, in
many instances there will be more than one
correct choice.)
A = B, A ⊆ B,
none of these
23) A = {2,4,6}
B ⊆ A, A ⊂ B, B ⊂ A, or
B = {2,4,6,8}
24) A = {2,5,7,9} B = {7,9}
Homework #19 – 34: Determine which of these
are true. (Choose every answer that is true, in
many instances there will be more than one
correct choice.)
A = B, A ⊆ B,
none of these
B ⊆ A, A ⊂ B, B ⊂ A, or
25) A = {a,b,c} B = {a,b,d}
26) A = {c,e,g,h} B = {c,b,a}
Homework #19 – 34: Determine which of these
are true. (Choose every answer that is true, in
many instances there will be more than one
correct choice.)
A = B, A ⊆ B,
none of these
B ⊆ A, A ⊂ B, B ⊂ A, or
27) A = {𝑥 |𝑥 ∈ 𝑁 𝑎𝑛𝑑 𝑥 < 9} B = {4,5,6}
28) A = {𝑥 |𝑥 ∈ 𝑁 𝑎𝑛𝑑 𝑥 < 5} B = {1,2,3}
Homework #19 – 34: Determine which of these
are true. (Choose every answer that is true, in
many instances there will be more than one
correct choice.)
A = B, A ⊆ B,
none of these
B ⊆ A, A ⊂ B, B ⊂ A, or
29) A = {8,9,10,11…}
B = {𝑥 |𝑥 ∈ 𝑁 𝑎𝑛𝑑 𝑥 ≥ 9}
30) A = {𝑥 |𝑥 ∈ 𝑁 𝑎𝑛𝑑 𝑥 ≥ 5} B = {4,5,6,7…}
Homework #19 – 34: Determine which of these
are true. (Choose every answer that is true, in
many instances there will be more than one
correct choice.)
A = B, A ⊆ B,
none of these
B ⊆ A, A ⊂ B, B ⊂ A, or
31) A = {𝑥 |𝑥 ∈ 𝑁 𝑎𝑛𝑑 2 < 𝑥 < 9}
B = {3,4,5,6,7,8}
32) A = {𝑥 |𝑥 ∈ 𝑁 𝑎𝑛𝑑 0 < 𝑥 < 5}
B = {1,2,3,4}
Homework #19 – 34: Determine which of these
are true. (Choose every answer that is true, in
many instances there will be more than one
correct choice.)
A = B, A ⊆ B,
none of these
B ⊆ A, A ⊂ B, B ⊂ A, or
33) A = {𝑥 |𝑥 ∈ 𝑁 𝑎𝑛𝑑 2 < 𝑥 < 9}
B = {0,1,2,3,4,5,6,7,8}
34) A = {𝑥 |𝑥 ∈ 𝑁 𝑎𝑛𝑑 0 < 𝑥 < 5}
B = {0,1,2,3,4}
#35-44: Answer true or false, if false give the
reason
35) 𝑐 ⊆ { a,b,c,d}
36) 2 ⊆ {2,3,4,5}
37)
{1,2,8} ⊆ {1,2,3,4,5}
38) {a,b,c} ⊆ {a,b,e,f,g}
39)
{1,2} ⊆ {1,2,3,4,5}
40) {a,b,c} ⊆ {a,b,e,f,g}
41) {1,2} ⊂ {1,2}
42) {𝑎, 𝑏, 𝑐} ⊂ {b,c,a}
43) {1,2,3} ⊂ {1,2,3,4}
44) {a,b,c,d} ⊂ {a,b,c,d,e}
Answers: 1) false, silver is not a set 3) true 5)
true 7) false “a” is not a set
9) true 11) true 13) false, they are equal and
the proper subset symbol does not allow sets
to be equal.
15) false, 5 is in the left set but not in the right
set 17) true
19) B ⊆ A, B⊂A 21) A = B, A ⊆ B, B⊆A 23) A
⊆ B, A ⊂ B 25) none
27) B ⊆ A, B⊂A 29) A ⊆ B, A ⊂ B 31) A =
B, A ⊆ B, B ⊆ A
33) A ⊆ B, A ⊂ B