Discrete mathematics, 6 rules of inference

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