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Conditional Statements & Implications – Mathematical Reasoning | Class 11 Maths

  • Difficulty Level : Hard
  • Last Updated : 19 Jan, 2021
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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”

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