Questions: Given A=p, q, r, s, t, B=p, q, C=p, q, w. Which of the following are true? There may be more than one correct answer. Select all that apply. Select all that apply: A ⊂ B B ⊂ C C ⊂ B A ⊂ C

Given A=p, q, r, s, t, B=p, q, C=p, q, w. Which of the following are true? There may be more than one correct answer. Select all that apply.

Select all that apply:
A ⊂ B
B ⊂ C
C ⊂ B
A ⊂ C

Transcript text: Given $A=\{\mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{s}, \mathrm{t}\}, B=\{\mathrm{p}, \mathrm{q}\}, C=\{\mathrm{p}, \mathrm{q}, \mathrm{w}\}$. Which of the following are true? There may be more than one correct answer. Select all that apply. Select all that apply: $A \subset B$ $B \subset C$ $C \subset B$ $A \subset C$

Solution

Solution Steps

To determine which of the given set relationships are true, we need to check if all elements of one set are contained within another set. Specifically, we will check the following:

If all elements of set A are in set B.
If all elements of set B are in set C.
If all elements of set C are in set B.
If all elements of set A are in set C.

Step 1: Define the Sets

We are given the sets: \[ A = \{ \text{p}, \text{s}, \text{r}, \text{q}, \text{t} \} \] \[ B = \{ \text{p}, \text{q} \} \] \[ C = \{ \text{p}, \text{w}, \text{q} \} \]

Step 2: Check if $ A \subset B $

To determine if $ A \subset B $, we check if all elements of $ A $ are in $ B $. Since $ A $ contains elements $\text{s}, \text{r}, \text{t}$ which are not in $ B $, we conclude: \[ A \not\subset B \]

Step 3: Check if $ B \subset C $

To determine if $ B \subset C $, we check if all elements of $ B $ are in $ C $. Since $ B $ contains elements $\text{p}, \text{q}$ which are in $ C $, we conclude: \[ B \subset C \]

Step 4: Check if $ C \subset B $

To determine if $ C \subset B $, we check if all elements of $ C $ are in $ B $. Since $ C $ contains the element $\text{w}$ which is not in $ B $, we conclude: \[ C \not\subset B \]

Step 5: Check if $ A \subset C $

To determine if $ A \subset C $, we check if all elements of $ A $ are in $ C $. Since $ A $ contains elements $\text{s}, \text{r}, \text{t}$ which are not in $ C $, we conclude: \[ A \not\subset C \]