Relevant definition:
“The set A is a subset of B if and only if every element of A is also an element of B. We use the notation A ⊆ B to indicate that A is a subset of the set B.” Discrete Mathematics and its Applications by Rosen.
To make sure that we check every possibility, we are going to make every possible combination of two sets. We can easily know the number of combinations using the theory that you will study later: Counting. This is an easy problem answered by the number of 2-permutations of 4 elements.
P(4,2)=4!/(4-2)!=12. You can find several step-by-step solutions in this link.
A⊆B?
4∈A and 4∉B
Therefore, A is not a subset of B.
A⊆C?
2∈A and 2∉C
Therefore, A is not a subset of C.
A⊆D?
6∈A and 6∉D
Therefore, A is not a subset of D.
B⊆A?
Every element of B is also an element of A. You can also write like this: ∀x∈B, x∈A.
Therefore, B is a subset of A.
B⊆C?
2∈B and 2∉C
Therefore, B is not a subset of C.
B⊆D?
2∈B and 2∉D
Therefore, B is not a subset of D.
C⊆A?
Every element of C is also an element of A. You can also write like this: ∀x∈C, x∈A.
Therefore, C is a subset of A.
C⊆B?
4∈C and 4∉B
Therefore, C is not a subset of B.
C⊆D?
Every element of C is also an element of D. You can also write like this: ∀x∈C, x∈D.
Therefore, C is a subset of D.
D⊆A?
8∈D and 8∉A.
Therefore, D is not a subset of A.
D⊆B?
8∈D and 8∉B.
Therefore, D is not a subset of B.
D⊆C?
8∈D and 8∉C
Therefore, D is not a subset of C.
Related exercises:
- Exercises 1 to 5 from the textbook
- 8. For each of the sets in Exercise 7, determine whether {2} is an element of that set
- 7. For each of the following sets, determine whether 2 is an element of that set
- 9. Determine whether each of these statements is true or false
- 10. Determine whether these statements are true or false
