Question

show step by step simplification: (¬𝑑∨((¬b∧c)∨(b∧¬c)))∧((𝑎 ∧ 𝑏) ∨ (¬𝑎 ∧ ¬𝑏))∧(¬𝑐∨((¬𝑑∧𝑎)∨(𝑑∧¬𝑎)))

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Miles

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55 Answers

To simplify the given expression, we can start by using the distributive law.

Step 1: Distribute the negation (¬) in the first part of the expression:

¬(d∨((¬b∧c)∨(b∧¬c)))

Step 2: Distribute the negation (¬) further inside the first part:

¬(d∨(¬b∧c)) ∧ ¬(d∨(b∧¬c))

Step 3: Apply De Morgan's laws and distribute the ∧ (logical AND) inside the first part:

(¬d∧(¬¬b∨¬c)) ∧ (¬d∧(¬b∨¬¬c))

Step 4: Simplify the double negations (¬¬):

(¬d∧(b∨¬c)) ∧ (¬d∧(¬b∨c))

Step 5: Apply the distributive law again to the second part of the expression:

((¬d∧b)∨(¬d∧¬c)) ∧ ((¬d∧¬b)∨(¬d∧c))

Step 6: Simplify the expressions with ∧ (logical AND):

((-d∧b)∨(-d∧¬c)) ∧ ((-d∧¬b)∨(-d∧c))

Step 7: Apply the distributive law once more:

(-d∧(b∨¬c))∨(-d∧(¬b∨c))

Step 8: Simplify the expressions with ∨ (logical OR):

(-d∧(b∨¬c))∨(-d∧(¬b∨c))

Step 9: Apply the distributive law to the ∨ (logical OR) outside the parentheses:

(-d∧b) ∨ ¬c ∨ (-d∧¬b) ∨ c

Step 10: Simplify the expressions with ∨ (logical OR) and ∧ (logical AND):

((-d∧b) ∨ (-d∧¬b)) ∨ ¬c ∨ c

Step 11: Simplify the expressions with ∧ (logical AND):

-d ∨ ¬c ∨ c

Step 12: Simplify the expressions with ∨ (logical OR):

-d ∨ 1

Answer:

-d ∨ 1

Step 1: Distribute the negation (¬) in the first part of the expression:

¬(d∨((¬b∧c)∨(b∧¬c)))

Step 2: Distribute the negation (¬) further inside the first part:

¬(d∨(¬b∧c)) ∧ ¬(d∨(b∧¬c))

Step 3: Apply De Morgan's laws and distribute the ∧ (logical AND) inside the first part:

(¬d∧(¬¬b∨¬c)) ∧ (¬d∧(¬b∨¬¬c))

Step 4: Simplify the double negations (¬¬):

(¬d∧(b∨¬c)) ∧ (¬d∧(¬b∨c))

Step 5: Apply the distributive law again to the second part of the expression:

((¬d∧b)∨(¬d∧¬c)) ∧ ((¬d∧¬b)∨(¬d∧c))

Step 6: Simplify the expressions with ∧ (logical AND):

((-d∧b)∨(-d∧¬c)) ∧ ((-d∧¬b)∨(-d∧c))

Step 7: Apply the distributive law once more:

(-d∧(b∨¬c))∨(-d∧(¬b∨c))

Step 8: Simplify the expressions with ∨ (logical OR):

(-d∧(b∨¬c))∨(-d∧(¬b∨c))

Step 9: Apply the distributive law to the ∨ (logical OR) outside the parentheses:

(-d∧b) ∨ ¬c ∨ (-d∧¬b) ∨ c

Step 10: Simplify the expressions with ∨ (logical OR) and ∧ (logical AND):

((-d∧b) ∨ (-d∧¬b)) ∨ ¬c ∨ c

Step 11: Simplify the expressions with ∧ (logical AND):

-d ∨ ¬c ∨ c

Step 12: Simplify the expressions with ∨ (logical OR):

-d ∨ 1

Answer:

-d ∨ 1

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