how to make negation of compound statement

In my last post we have seen duality in logic and some of its examples.

In this article  we are going to learn negation of compound statements and some of its examples. So let’s begin.

1) Negation of Conjunction.

We have seen in my previous article on logical equivalence that    ~ (p Ʌ q) ≡ ~ p V ~q by De Morgan’s Laws

Ex.  Sky is blue and milk is white.

Let p: sky is blue   and q: milk is white.

Then given statement in symbolic form is p Ʌ q.

Its negation is ~ p V ~q

i.e. Sky is not blue or milk is not white.

2) Negation of Dis junction.

We have seen in my previous post on logical equivalence that     ~ (p V q) ≡ ~ p Ʌ ~q by De Morgan’s Laws

Ex.  Sky is blue or 6>5.

Let p: sky is blue   and q: 6>5.

Then given statement in symbolic form is p V q.

Its negation is   ~ p Ʌ ~q.

i.e. Sky is not blue and 6＜or= 5.

3) Negation of Negation.

We know that ~(~p) ≡p

e. g. p: sky is blue.

   ~p is ‘Sky is not blue’ 

And ~ (~p is ‘It is false that sky is not blue’.

Or p: sky is blue.

4) Negation of Implication.

We have seen in my previous post on logical equivalence that     p → q ≡   ~p V q

Now ~ (p → q) ≡ ~ (~p V q)

                           ≡ ~(~p) Ʌ ~q     By negation of dis-junction

                           ≡ p Ʌ ~q             by negation of negation

Therefore ~ (p → q) ≡ p Ʌ ~q

Ex. If sky is blue then milk is white.

Let p: sky is blue    and q: milk is white.

Its symbolic form is p → q.

Its negation is p Ʌ ~q

i.e. Sky is blue and milk is not white.

5) Negation of Bi-conditional.

We have seen in my previous post on logical equivalence that p ↔ q ≡ (p → q) Ʌ (q → p)

~ (p ↔ q) ≡ ~ [(p → q) Ʌ (q → p)]

                   ≡ ~ (p → q) V ~ (q → p) by negation of conjunction

                    ≡ (p Ʌ ~q) V (q Ʌ ~p) by negation of implication

Therefore ~ (p ↔ q) ≡ (p Ʌ ~q) V (q Ʌ ~p)

e.g. Sky is blue if and only if milk is white.

Let p: sky is blue    and q: milk is white.

Its symbolic form is p↔ q.

Its negation is (p Ʌ ~q) V (q Ʌ ~p)

i.e. Sky is blue and milk is not white or milk is white and Sky is not blue.

6) Negation of quantified statements.

While doing negations of quantified statements, we replace word ‘all’ by ‘some’, “for every” by “there exits” (ꓯ by ⱻ) and vice versa and add the word ‘not’ at the appropriate place.

Ex.  1) All natural numbers are real.

→ Some natural numbers are not real.

2) ꓯ n Є N, n+ 7＞6

→ ⱻ n Є N, such that n+ 7＜or= 6

This is all about negation of compound statement.

In my next post we are going to learn about application of logic to the switching circuits.