Question
Exercise
Let the following variables be defined:
x=10, y=15, z=20.
For each expression below, determine whether it evaluates to True or False.
1. (x y)
2. (x+y = z) OR (x×2 = z)
3. (y−x
4. NOT (y+5
5. (z/y≥1) AND (z−x≤y+5)
6. (x+y≤z×2) OR (z−y==x)
Your task: Evaluate each compound expression step by step. Decide if each part of the expression is true or false, and then combine the results using the specified Boolean operators (AND, OR, NOT).
Let the following variables be defined:
x=10, y=15, z=20.
For each expression below, determine whether it evaluates to True or False.
1. (x y)
2. (x+y = z) OR (x×2 = z)
3. (y−x
4. NOT (y+5
5. (z/y≥1) AND (z−x≤y+5)
6. (x+y≤z×2) OR (z−y==x)
Your task: Evaluate each compound expression step by step. Decide if each part of the expression is true or false, and then combine the results using the specified Boolean operators (AND, OR, NOT).
Answers
GPT-4o mini
Let's evaluate each expression step by step based on the defined values:
- \( x = 10 \)
- \( y = 15 \)
- \( z = 20 \)
1. **Expression:** \( (x \, y) \)
This expression is invalid as it doesn't have a logical operator between \( x \) and \( y \).
2. **Expression:** \( (x + y = z) \, \text{OR} \, (x \times 2 = z) \)
- Evaluate \( (x + y = z) \):
\( 10 + 15 = 25 \), which is **False**.
- Evaluate \( (x \times 2 = z) \):
\( 10 \times 2 = 20 \), which is **True**.
- Combine using OR:
\( \text{False} \, \text{OR} \, \text{True} = \text{True} \).
3. **Expression:** \( (y - x) \)
- Evaluate \( (y - x) \):
\( 15 - 10 = 5 \), which is a value, not a Boolean expression. However, if we consider non-zero as **True**: **True**.
4. **Expression:** \( \text{NOT} \, (y + 5) \)
- Evaluate \( (y + 5) \):
\( 15 + 5 = 20 \), which is a positive value. Under most Boolean contexts, positive numbers are treated as **True**.
- Apply NOT:
\( \text{NOT True} = \text{False} \).
5. **Expression:** \( (z/y \geq 1) \, \text{AND} \, (z - x \leq y + 5) \)
- Evaluate \( (z/y \geq 1) \):
\( 20/15 = 1.33 \), which is **True** since \( 1.33 \geq 1 \).
- Evaluate \( (z - x \leq y + 5) \):
\( 20 - 10 = 10 \) and \( y + 5 = 15 + 5 = 20 \). So, \( 10 \leq 20 \) is **True**.
- Combine using AND:
\( \text{True} \, \text{AND} \, \text{True} = \text{True} \).
6. **Expression:** \( (x + y \leq z \times 2) \, \text{OR} \, (z - y == x) \)
- Evaluate \( (x + y \leq z \times 2) \):
\( 10 + 15 = 25 \) and \( z \times 2 = 20 \times 2 = 40 \). So, \( 25 \leq 40 \) is **True**.
- Evaluate \( (z - y == x) \):
\( 20 - 15 = 5 \), and \( 5 == 10 \) is **False**.
- Combine using OR:
\( \text{True} \, \text{OR} \, \text{False} = \text{True} \).
### Summary of Evaluations:
1. Invalid expression (not evaluated).
2. True
3. True
4. False
5. True
6. True
If there's anything else you'd like to clarify or further explore, let me know!
- \( x = 10 \)
- \( y = 15 \)
- \( z = 20 \)
1. **Expression:** \( (x \, y) \)
This expression is invalid as it doesn't have a logical operator between \( x \) and \( y \).
2. **Expression:** \( (x + y = z) \, \text{OR} \, (x \times 2 = z) \)
- Evaluate \( (x + y = z) \):
\( 10 + 15 = 25 \), which is **False**.
- Evaluate \( (x \times 2 = z) \):
\( 10 \times 2 = 20 \), which is **True**.
- Combine using OR:
\( \text{False} \, \text{OR} \, \text{True} = \text{True} \).
3. **Expression:** \( (y - x) \)
- Evaluate \( (y - x) \):
\( 15 - 10 = 5 \), which is a value, not a Boolean expression. However, if we consider non-zero as **True**: **True**.
4. **Expression:** \( \text{NOT} \, (y + 5) \)
- Evaluate \( (y + 5) \):
\( 15 + 5 = 20 \), which is a positive value. Under most Boolean contexts, positive numbers are treated as **True**.
- Apply NOT:
\( \text{NOT True} = \text{False} \).
5. **Expression:** \( (z/y \geq 1) \, \text{AND} \, (z - x \leq y + 5) \)
- Evaluate \( (z/y \geq 1) \):
\( 20/15 = 1.33 \), which is **True** since \( 1.33 \geq 1 \).
- Evaluate \( (z - x \leq y + 5) \):
\( 20 - 10 = 10 \) and \( y + 5 = 15 + 5 = 20 \). So, \( 10 \leq 20 \) is **True**.
- Combine using AND:
\( \text{True} \, \text{AND} \, \text{True} = \text{True} \).
6. **Expression:** \( (x + y \leq z \times 2) \, \text{OR} \, (z - y == x) \)
- Evaluate \( (x + y \leq z \times 2) \):
\( 10 + 15 = 25 \) and \( z \times 2 = 20 \times 2 = 40 \). So, \( 25 \leq 40 \) is **True**.
- Evaluate \( (z - y == x) \):
\( 20 - 15 = 5 \), and \( 5 == 10 \) is **False**.
- Combine using OR:
\( \text{True} \, \text{OR} \, \text{False} = \text{True} \).
### Summary of Evaluations:
1. Invalid expression (not evaluated).
2. True
3. True
4. False
5. True
6. True
If there's anything else you'd like to clarify or further explore, let me know!