Sure, let's analyze each of the given statements step-by-step to determine whether they are true or false based on the sets provided:
Given sets:
[tex]\[ P = \{0, 2, 4\} \][/tex]
[tex]\[ Q = \{x \mid x \text{ is an odd number}\} \][/tex]
[tex]\[ R = \{2\} \][/tex]
### a. [tex]\( P \subseteq Q \)[/tex] ?
We need to determine if [tex]\( P \)[/tex] is a subset of [tex]\( Q \)[/tex], meaning every element in [tex]\( P \)[/tex] should also be in [tex]\( Q \)[/tex].
- [tex]\( P \)[/tex] contains the elements \{0, 2, 4\}.
- [tex]\( Q \)[/tex] is the set of all odd numbers.
Since [tex]\( P \)[/tex] contains the elements 0, 2, and 4, and none of these elements are odd (Q contains elements like -3, -1, 1, 3, etc.), none of the elements in [tex]\( P \)[/tex] are in [tex]\( Q \)[/tex].
Therefore:
[tex]\[ P \subseteq Q \][/tex] is [tex]\(\text{False}\)[/tex].
### b. [tex]\( R \subseteq P \)[/tex] ?
We need to determine if [tex]\( R \)[/tex] is a subset of [tex]\( P \)[/tex], meaning every element in [tex]\( R \)[/tex] should also be in [tex]\( P \)[/tex].
- [tex]\( R \)[/tex] contains the single element \{2\}.
- [tex]\( P \)[/tex] contains the elements \{0, 2, 4\}.
The element 2 is indeed an element of [tex]\( P \)[/tex].
Therefore:
[tex]\[ R \subseteq P \][/tex] is [tex]\(\text{True}\)[/tex].
### c. [tex]\( R \subseteq Q \)[/tex] ?
We need to determine if [tex]\( R \)[/tex] is a subset of [tex]\( Q \)[/tex], meaning every element in [tex]\( R \)[/tex] should also be in [tex]\( Q \)[/tex].
- [tex]\( R \)[/tex] contains the single element \{2\}.
- [tex]\( Q \)[/tex] is the set of all odd numbers.
Since 2 is not an odd number, it is not an element of [tex]\( Q \)[/tex].
Therefore:
[tex]\[ R \subseteq Q \][/tex] is [tex]\(\text{False}\)[/tex].
### d. [tex]\( R \nsubseteq Q \)[/tex] ?
We need to determine if [tex]\( R \)[/tex] is not a subset of [tex]\( Q \)[/tex], meaning not all elements in [tex]\( R \)[/tex] are in [tex]\( Q \)[/tex].
From the previous part, we've established that [tex]\( R \subseteq Q \)[/tex] is false because 2 is not an element of [tex]\( Q \)[/tex]. Hence, it directly implies that [tex]\( R \)[/tex] is not a subset of [tex]\( Q \)[/tex].
Therefore:
[tex]\[ R \nsubseteq Q \][/tex] is [tex]\(\text{True}\)[/tex].
### Summary
The answers for the statements are:
- a. [tex]\( P \subseteq Q \)[/tex]: [tex]\(\text{False}\)[/tex]
- b. [tex]\( R \subseteq P \)[/tex]: [tex]\(\text{True}\)[/tex]
- c. [tex]\( R \subseteq Q \)[/tex]: [tex]\(\text{False}\)[/tex]
- d. [tex]\( R \nsubseteq Q \)[/tex]: [tex]\(\text{True}\)[/tex]
