The document outlines 6 rules of inference used to derive logical conclusions: 1) Conjunctive simplification, 2) Disjunctive amplification, 3) Hypothetical syllogism, 4) Disjunctive syllogism, 5) Modus ponens, and 6) Modus tollens. Each rule is symbolically represented and includes an explanation of its application using truth tables to prove the validity of the logical inferences.

1. RULES OF INFERENCE (CONCLUSION)
1. Rule of conjunctive simplification
This rule states that, P is true whenever PΛQ is true.
Symbolically it is PΛQ
∴P
Explanation: pΛq is similar to AND logic of Digital electronics. Its truth value is 1 only if
p and q both are 1. So we can say p is true whenever PΛQ is true.
Proof by truth table:
P Q PΛQ
0 0 0
0 1 0
1 0 0
1 1 1
2. Rule of disjunctive amplification
This rule states that P⋁Q is true whenever P is true.
Symbolically it is P
∴ P⋁Q
Explanation: P⋁Q is similar to OR logic of Digital electronics. Its truth value is 1 if any of
the inputs ( p or q )is 1. So we can say P⋁Q is true whenever P is true, irrespective of
the value of Q.
Proof By Truthtable:
P Q PVQ
0 0 0
0 1 1
1 0 1
1 1 1

2. 3. Rule of Hypothetical syllogism:
This rule states that P⟶R is true whenever P⟶ Q is true and Q⟶R is true.
Symbolically it is P⟶Q
Q⟶R
∴ P⟶R
Explanation: P⟶Q is true, it states that P is not 1 and Q is not 0 concurrently.
Same way, Q⟶R is true, it states that Q is not 1 and R is not 0 at the same time.
(Because A⟶B is a conditional statement whose value is 0 only in one case which is
when A is1 and B is 0)So we can say both these statements can take any other
combination other than 1 and 0.
Let’s say P is 1 ie Q is also 1.
Going to statement (b), if Q is 1 from above statement, R must be 1.
And if P is 1 and Q is 1 and R is also 1. This implies P⟶R is also one that is only our
conclusion.
Proof By Truthtable
P Q R P⟶Q Q⟶R P⟶R
0 0 0 1 1 1
0 0 1 1 1 1
0 1 0 1 0 0
0 1 1 1 1 1
1 0 0 0 1 1
1 0 1 0 1 1
1 1 0 1 0 0
1 1 1 1 1 1

3. 4. Rule of Disjunctive syllogism
This rule states that Q is true whenever P⋁Q is true and P is true.
Symbolically it is P⋁Q (a)
P (b)
∴Q
Explanation: P⋁Q behaves the same way as OR gate in digital electronics ie, if any of the
two inputs is 1 the output is 1. So if we know that PVQ is 1(from (a)) and P is 1 that
means P is 0. So to support (a), Q has to be one only then the output of PVQ will be 1.
This implies Q is 1 and that is our conclusion.
Proof By Truthtable
P Q P⋁Q P
0 0 0 1
0 1 1 1
1 0 1 0
1 1 1 0
5. Rule of Modus Pones( Rule of detachment)
This rule states that Q is true whenever P is true and P⟶Q is true.
Symbolically it is P (a)
P⟶Q (b)
∴Q
Explanation: P⟶Q is a conditional statement in discrete mathematics, which is false
only in one case that is when p is true and q is false (or P is 1 and Q is 0 then p⟶q is 0
and in all other cases it is true.)
Here we know that p is true, from (a)
And P⟶Q is true (from (b)) ie Q must be true that is the conclution.

4. Proof By Truthtable
P Q P⟶Q
0 0 1
0 1 1
1 0 0
1 1 1
6. Rule of Modus Tollens
This rule states that ~P is true whenever P⟶Q is true and ~Q is true.
Symbolically it is P⟶Q ……………… (a)
~Q ……………….. (b)
∴~P
Explanation: P⟶Q is a conditional statement in discrete mathematics, which is false
only in one case that is when p is true and q is false (or P is 1 and Q is 0 then p⟶q is is
0 and in all other cases it is true)
We know that ~Q is 1from (b) ie Q is 0 ( we know that ~Q is inverse of Q)
The above condition P⟶Q can be true only if P is 0 because if P is 1 and Q is 0 the
output will be 0.
So it signifies that p is 0 and ~P is 1 that is our conclusion.
Proof By Truthtable
P Q ~Q ~P P⟶Q
0 0 1 1 1
0 1 0 1 1
1 0 1 0 0
1 1 0 0 1