Biconditionals

Biconditionals

Lessons

A biconditional is a conjunction of a conditional and its converse. In symbolic form it would be:

(pq)(p \to q) ˄ (qp)(q \to p)

We can also write it as pqp \leftrightarrow q. In words, we connect pp and qq with "if and only if". A biconditional can be only true if both the conditional and converse is true. Here is a truth table of a biconditional.

pp qq pqp \to q qpq \to p (pq)(p \to q ) ˄ (qp)(q \to p) pq p \leftrightarrow q
T T T T T T
T F F T F F
F T T F F F
F F T T T T
  • Introduction
    Biconditionals Overview:
    a)
    What are Biconditionals


  • 1.
    Forming the Biconditional
    Write a biconditional using the given conditionals:
    a)
    If today is Monday, then yesterday was Sunday.
    If yesterday was Sunday, then today is Monday

    b)
    If he drives faster than 100 km, then he will get a ticket.
    If he gets a ticket, then he has driven faster than 100 km.

    c)
    If today is the weekend, then I do not go to school.
    If I do not go to school, then it is the weekend.


  • 2.
    You are given 3 statements in symbolic form:

    pp: There is cream in milk
    qq: There is milk in coffee
    rr: There is cream in coffee

    a)
    Write the conditional (p (p ˄ q)q) \to rr in words.

    b)
    Write the converse(p (p ˄ q)q) \to rr in words.

    c)
    Write the biconditional.


  • 3.
    Truth value of Biconditionals
    Determine the truth value of the following biconditionals:
    a)
    x=5x = 5. if and only if x+7=12x + 7 = 12.

    b)
    If x>5x > 5, if and only if x>2x > 2 .

    c)
    The triangle is an isosceles, if and only if two sides are equal.

    d)
    The angle is acute, if and only if the angle is less than 90° degrees.


  • 4.
    Let both pp, qq and rr be statements.
    a)
    If pp is true, qq is true and rr is false, then what is the truth value of the biconditional (p(p ˄ q)q) \leftrightarrowrr ?

    b)
    If pp is false, qq is true and rr is false, then what is the truth value of the biconditional (p(p ˄ q)q) \leftrightarrowrr ?

    c)
    If pp is false, qq is false and rr is true, then what is the truth value of the biconditional (p(p ˄ q)q) \leftrightarrowrr ?

    d)
    If pp is false, qq is false and rr is false, then what is the truth value of the biconditional (p(p ˄ q)q) \leftrightarrowrr ?