### Proposition :-

A simple proposition ‘P’ is a linguistic statement contained within a universe of elements ‘x’ that can be identified as being a collection of elements in ‘x’ that are simple true or simply false.

### Truthness or Veracity of proposition P :-

Truthness (or Veracity) is denoted by T(P) and it can be assigned a binary truth value.

Let P and Q be two simple propositions defined on same universe of discourse, these simple propositions can be combined using the following 5 logical connectives:

### Disjunction (V) :-

Let the two logical propositions be

- P : truth that x ∈ A
- Q : truth that x ∈ B

T(P) = 1 when x ∈ A otherwise T(P) = 0

T(Q) = 1 when x ∈ B otherwise T(Q) = 0

P V Q = x ∈ A or x ∈ B

hence T (P V Q) = T(P) V T(Q) = max [ T(P), T(Q)]

### Conjunction :-

P / Q = x ∈ A and x ∈ B

hence T(P / Q) = min [ T(P), T(Q)]

### Negations :-

If T(P) = 0 then T(P^{c}) = 1

T(P) = 1 then T(P^{c}) = 0

### Implication (→) :-

P → Q = x ∉ A or x ∈ B

T(P → Q) is true ∀ cases except when,

first proposition T(P) = 1 is true and

second proposition T(Q) = 0 is false

Hence T(P → Q) = T (P^{c} V Q)

The logical connective application that is P → Q (P implies Q) is true in all cases except when P is true and Q i false.

In P → Q the simple propositions P and Q are called as below:

- P is called hypothesis or anticident.
- Q is called consequence or conclusion.

So finally, “A true anticident cannot imply a false consequence.”

or it can also be said as – “A true hypothesis cannot imply a false conclusion.”

### Equivalence (↔) :-

When P ↔ Q it is called compound proposition.

T(P ↔ Q) = 1 for T(P) = T(Q)

T(P ↔ Q) = 0 for T(P) ≠ T(Q)

T(P) | T(Q) | T(P V Q) | T(P / Q) | T (P^{c} | T(P → Q) | T(P ↔ Q) |

0 | 0 | 0 | 0 | 1 | 1 | 1 |

0 | 1 | 1 | 0 | 1 | 1 | 0 |

1 | 0 | 1 | 0 | 0 | 0 | 0 |

1 | 1 | 1 | 1 | 0 | 1 | 1 |