Propositional Logic

A statement is a declarative statement that is either true or false, never both. A statement can either be primitive or compound.

• Primitive statement - India is a nice place to live.
• Compound statement - India is a nice place to live and people in India are really friendly.

AND $(\wedge )$, OR $(\vee )$, and NEGATION $(\neg )$ are called the primitive/basic logical operators. Their truth tables isn’t worth discussing.

Implication

The implication operator $⟹$ in any statement like $A⟹B$ means that if $B$ is true, then $B$ must be true. It doesn’t mean that $A$ and $B$ cause each other, it just means that $A$ being true cannot coexist with $B$ being false.

The truth table of the implication operator looks like -

$A$	$B$	$A⟹B$
0	0	1
0	1	1
1	0	0
1	1	1

The truth table can seem confusing as the implication is considered true even when $A$ is false. This is because it just means that $A$ being true cannot coexist with $B$ being false. When $A$ is false, the implication is vacuously (by default) considered true.

For any statement $A⟹B$ to be an implication, its truth table should be the same as an implication’s.

Double Implication

When for two statements $A$ and $B$, $A⟹B$ and $B⟹A$, we say that $A⟺B$. As we are looking at a two-way implication here, the truth table of double implication will only be false when one statement is true and the other isn’t.

$A$	$B$	$A⟹B$
0	0	1
0	1	0
1	0	0
1	1	1

Converse, Inverse, and Contrapositive

The implication at hand is $p⟹q$ -

1. Converse of this implication will be $q⟹p$.
2. Inverse of this implication will be $\neg p⟹\neg q$.
3. Contrapositive of this implication will be $\neg q⟹\neg p$.