Conditional Statements & Implications – Mathematical Reasoning | Class 11 Maths

Generally, Conditional statements are the if-then statement in which p is called a hypothesis(or antecedent or premise) and q is called a conclusion( or consequence). Conditional Statements symbolized by p, q. A Conditional statement p -> q is false when p is true and q is false, and true otherwise. 

What are propositions?

A proposition is a declarative statement that is either true or false, but not both.

Examples:

• Delhi is the capital of India
• 1 + 1 = 2
• 2 + 2 = 4

Let p and q are propositions. 

• The conditional statement p -> q is the proposition “if p, then q”. 
• The conditional statement p -> q is false when p is true and q is false and true in all other cases.

By the following table, we can identify the values of implications:

p	q	p -> q
T	T	T
T	F	F
F	T	T
F	F	T

Variety of terminology is used to express p -> q

• ” if p then q
• “if p, q “
• “q if p”
• “q when p”
• “q unless p”
• “p implies q”
• “p only if q”
• “q whenever p”
• “q follows from p”

Conditional statements are also called implications. The statement is an implication p -> q is called its hypothesis, and q the conclusion.

Example: Let p be the statement “Maria learn Java Programming ” and q is the statement “Maria will find a good job”. Express the statement p -> q as a statement in English?

Solution: 

“If Maria learns java programming, then she will find a good job”.

                                                                             or

“Maria will find a good job when she learns java programming.”

Converse, Contrapositive, and Inverse

We can form some new conditional statements starting with a conditional statement p -> q. 

1. The converse of p -> q is the proposition q -> p.
2. The contrapositive of p -> q is the proposition ~q -> ~p.
3. The inverse of p -> q is the proposition ~p -> ~q.

By the following table, we can identify the values of Converse, Contrapositive, and Inverse:

p	q	~p	~q	p -> q	~q -> ~p
T	T	F	F	T	T
T	F	F	T	F	F
F	T	T	F	T	T
F	F	T	T	T	T

Note: The contrapositive always has the same truth value as p -> q. When two compound propositions always have the same truth value we call them equivalent, so conditional statement and its contrapositive are equivalent. The converse and the inverse of a conditional statement are also equivalent.

Example 1: Show that p -> q and its contrapositive ~q -> ~p are logically equivalent.

Solution:

p	q	~p	~q	p -> q	~q -> ~p
T	T	F	F	T	T
T	F	F	T	F	F
F	T	T	F	T	T
F	F	T	T	T	T

As p ->q is equal to ~q -> ~p, hence both propositions are equivalent. 

Example 2: Show that proposition q -> p, and ~p -> ~q is not equivalent to p -> q.

Solution:

p	q	~p	~q	p -> q	q -> p	~p -> ~q
T	T	F	F	T	T	T
T	F	F	T	F	T	T
F	T	T	F	T	F	F
F	F	T	T	T	T	T

In this case, p -> q is not equal to q -> p and ~p -> ~q, hence they are not equal to p -> q but they themselves are equal.

Example 3: What is contrapositive, the converse, and the inverse of the conditional statement “The home team wins whenever it is raining.”?

Solution:

Because “q whenever p” is one way to express conditional statements p -> q.

Original sentence:

“If it is raining, then the home team wins”.  

• Contrapositive: “If the home team does not win, then it is not raining.”
• Converse: “If the home team wins, then it is raining.”
• Inverse: “If it is not raining, then the home team does not win.”

Example 4: What are contrapositive, the converse, and the inverse of the conditional statement “If the picture is a triangle, then it has three sides.”?

Solution:

• Contrapositive: “If the picture doesn’t have three sides, then it is not a triangle.”
• Converse: “If the picture has three sides, then it is a triangle.”
• Inverse: “If the picture is not a triangle, then it doesn’t have three sides.”

Biconditional or Equivalence

• We now introduce another way to combine propositions that express that two propositions have the same truth values.
• Let p and q be propositions.
• The biconditional statement p <-> q is the propositions “p if and only if q”
• The biconditional statement p <-> q is true when p and q have the same truth values and is false otherwise.
• Biconditional statements are also called bi-implications.
• There are some common way to express p<->q
• “p is necessary and sufficient for q”
• “if p then q,  and conversely”
• “p if q”.

By the following table, we can identify the values of Biconditional:

p	q	p <-> q
T	T	T
T	F	F
F	T	F
F	F	T

Example: What is the Biconditional of these following sentences. Let p be the statement” You can take the flight” and let q be the statement “You buy a ticket.”

Solution:

p <-> q is  “You can take the flight if and only if you buy a ticket”