[(m→¬n)∨(m˄n)]˄p
Step-by-Step Simplification Table:
Step	Expression	Rule Applied
1	
[
(
𝑚
→
¬
𝑛
)
∨
(
𝑚
˄
𝑛
)
]
˄
𝑝
[(m→¬n)∨(m˄n)]˄p	Original Expression
2	
[
(
¬
𝑚
∨
¬
𝑛
)
∨
(
𝑚
˄
𝑛
)
]
˄
𝑝
[(¬m∨¬n)∨(m˄n)]˄p	Implication Rule: 
𝑚
→
¬
𝑛
=
¬
𝑚
∨
¬
𝑛
m→¬n=¬m∨¬n
3	
(
¬
𝑚
∨
¬
𝑛
∨
𝑚
˄
𝑛
)
˄
𝑝
(¬m∨¬n∨m˄n)˄p	Associative Law: Combine 
(
¬
𝑚
∨
¬
𝑛
)
(¬m∨¬n) and 
(
𝑚
˄
𝑛
)
(m˄n)
4	
(
¬
𝑚
∨
¬
𝑛
∨
𝑚
)
˄
(
¬
𝑚
∨
¬
𝑛
∨
𝑛
)
˄
𝑝
(¬m∨¬n∨m)˄(¬m∨¬n∨n)˄p	Distributive Law: Expand 
¬
𝑚
∨
¬
𝑛
∨
(
𝑚
˄
𝑛
)
¬m∨¬n∨(m˄n)
5	
(
True
)
˄
(
True
)
˄
𝑝
(True)˄(True)˄p	Complement Law: 
¬
𝑚
∨
𝑚
=
True
,
 
¬
𝑛
∨
𝑛
=
True
¬m∨m=True, ¬n∨n=True
6	
𝑝
p	Identity Law: 
True˄
𝑝
=
𝑝
True˄p=p


Let's go through the step-by-step simplification table to ensure clarity and correctness in each step:

### 1. Original Expression:
\[
[(m \to \neg n) \vee (m \wedge n)] \wedge p
\]
This is the starting point, and we aim to simplify this expression.

### 2. Apply the Implication Rule:
The implication \( m \to \neg n \) can be rewritten using the rule \( m \to \neg n = \neg m \vee \neg n \):
\[
[(\neg m \vee \neg n) \vee (m \wedge n)] \wedge p
\]
This is the correct transformation from the original implication.

### 3. Apply the Associative Law:
We now combine the terms using the associative law of disjunction. The expression becomes:
\[
(\neg m \vee \neg n \vee (m \wedge n)) \wedge p
\]
We don't do much simplifying yet here; this step is primarily about grouping the terms.

### 4. Apply the Distributive Law:
Next, we distribute \( \neg m \vee \neg n \) over the conjunction \( m \wedge n \). This step involves expanding:
\[
(\neg m \vee \neg n \vee m) \wedge (\neg m \vee \neg n \vee n) \wedge p
\]
Here, we break up the disjunction \( \neg m \vee \neg n \vee (m \wedge n) \) into two parts as a distributive step. The logic is:
- \( (\neg m \vee \neg n \vee m) \) represents the first part, and
- \( (\neg m \vee \neg n \vee n) \) represents the second part.

### 5. Apply the Complement Law:
Now, we apply the complement law. We know that:
- \( \neg m \vee m = \text{True} \),
- \( \neg n \vee n = \text{True} \).

Thus, the expression simplifies to:
\[
\text{True} \wedge \text{True} \wedge p
\]

### 6. Apply the Identity Law:
Finally, applying the identity law, \( \text{True} \wedge p = p \), the expression simplifies to:
\[
p
\]

### Final Result:
The simplified expression is \( p \).

---

Would you like further details or have any questions on the steps?

---

Here are five related questions you might find interesting:
1. How does the distributive

Explore how to simplify the logical expression [(m→¬n)∨(m˄n)]˄p using key logical laws, including the implication rule, distributive law, and complement law. This step-by-step guide helps high school students understand logical operations and simplifications. Suitable for students in Grades 10-12.

Math Problem Statement

True , ¬ 𝑛 ∨ 𝑛

True ¬m∨m=True, ¬n∨n=True 6 𝑝 p Identity Law: True˄ 𝑝

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation