¬(p ∧ q) ≡ ¬p ∨ ¬q¬(p ∨ q) ≡ ¬p ∧ ¬q
To prove the equivalence of the given logical statements, we will use the laws of De Morgan’s theorem and the distributive law
To prove the equivalence of the given logical statements, we will use the laws of De Morgan’s theorem and the distributive law. Let’s start with the first statement:
¬(p ∧ q) ≡ ¬p ∨ ¬q
To prove this, we will work on both sides.
Starting with the left-hand side (LHS):
¬(p ∧ q)
By applying De Morgan’s theorem to the inner part of the expression:
¬p ∨ ¬q
This is the same as the right-hand side (RHS) of the equation.
Therefore, we have proved that ¬(p ∧ q) ≡ ¬p ∨ ¬q.
Now, let’s move on to the second statement:
¬(p ∨ q) ≡ ¬p ∧ ¬q
Again, we will work on both sides to prove the equivalence.
Starting with the LHS:
¬(p ∨ q)
By applying De Morgan’s theorem to the inner part of the expression:
¬p ∧ ¬q
This is the same as the RHS of the equation.
Therefore, we have proved that ¬(p ∨ q) ≡ ¬p ∧ ¬q.
In conclusion, we have proved both logical equivalences:
¬(p ∧ q) ≡ ¬p ∨ ¬q
¬(p ∨ q) ≡ ¬p ∧ ¬q
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