Validating statements in mathematical reasoning

Validating the statements involving the connecting words difference between contradiction, converse and contrapositive

Rule 1: If p and q are mathematical statements, then to show that ‘p and q’ are true, follow the steps below:

1. Show that p is a true statement
2. Show that q is a true statement

For statements with ‘or’

Rule 2: If p and q are mathematical statements, then to show that p and q are true statements, follow the steps below:

1. Assuming that p is a false statement, show that q must be true.
2. Assuming that q is false, show that p must be true.

For statements with ‘if-then’

Rule 3: To prove the if p then q statements, we need to verify that any one of the following cases must be true

1. Assuming that p is true, use the direct method to show that q is true
2. Assuming that q is false, use the contrapositive method to show that p must be false

For statements with “if and only if”

Rule 4: To prove that the statement“ p if and only if q” is valid, we need to show that

1. If p is true, then q is true and
2. If q is true, then p is also true
3. If p is false then q is also false and vice verse

A conditional statement has two parts to understand, these are a hypothesis and a conclusion.

When the if-then form is used to write a conditional statement, the part after the “if” is the hypothesis, and the part after the “then” is the conclusion. It is written as p → q

The converse of a conditional statement can be obtained by exchanging the hypothesis and the conclusion.The converse of p → q is written as q → p

The inverse is the statement that we obtain by negating the hypothesis and conclusion: ~p ~q

A contradiction, c is a statement form that is false regardless of the truth values of the statement variables. The statement for p Λ ~ p is a contradiction.

A contrapositive negates the hypothesis and the conclusion of the converse. The contrapositive of p → q is written as ~q → ~p.

When any two statements together are either both true or both false

• A conditional statement is equivalent to its contrapositive.
• The inverse and the converse of any conditional are equivalent.

Below are the truth values of the related statements above. Equivalent statements have the same truth value.

Example: If an animal is a cat, then it has four paws.

Converse: If an animal has 4 paws, then it is a lion.

There are other animals that have 4 paws that are not lions, so the converse is false.

Inverse: If an animal is not a lion, then it does not have 4 paws.

There are animals that are not the lion that have 4 paws, so the inverse is false.

Contrapositive: If an animal does not have 4 paws, then it is not a lion; True. Lions have 4 paws, so the contrapositive is true.