## Chapter - 1, Sentential Logic

### Section - 1.5 - The Conditional and Biconditional Connectives

This post contains solutions of Chapter - 1, Section - 1.5, The Conditional and Biconditional Connectives from Velleman’s book **How To Prove It**.

### Summary

- is equivalent to :
- .
- .

- and are not equivalent. They are called converse of each other.
- and are equivalent. They are called contrapositive of each other.
- is equivalent to following:
- implies .
- , if .
- , only if .
- is a sufficient condition for .
- is necessary condition for .

- is equivalent to . Thus it means:
- is a necessary and sufficient condition for .
- .

### Solutions

**Soln1**

**(a)**

Gas has pleasant smell.

Gas is explosive.
Gas is hydrogen.

. This is equivalent to which is equivalent to .

**(b)**

Fever, Headache, Doctor.

.

**(c)** .

**(d)** , where is “x is prime” and is “x is odd”.

**Soln2**

**(a)**

Good Price, Nice Apartment, Sell house.

.

**(b)**

Good Credit History, Adequate down payment, Getting mortgage.

.

**(c)**

If someone not stops John then he will kill himself. .

**(d)** , where means “ x is divisible by y”, means x is prime.

**Soln3**

Raining, Windy, Shining.

**(a)** .

**(b)** . It is converse of (a).

**(c)** . It is equivalent to (a).

**(d)** . It is converse of (a).

**(e)** . It is same as . It is equivalent to (a).

**(f)** . It is equivalent to . It is equivalent to (a).

**(g)** . IT is equivalent to . It is converse of (a).

**Soln4**

**(a)**

true | true | true | true | true | false | false |

true | true | false | true | false | true | false |

true | false | true | true | true | true | true |

true | false | false | true | false | true | true |

false | true | true | true | true | false | true |

false | true | false | true | true | true | true |

false | false | true | false | true | true | true |

false | false | false | false | true | true | true |

From the table when all premises : are true then conclusion, is also true.

**(b)**

true | true | true | true | true | true | false | true | false |

true | true | true | false | true | false | true | true | false |

true | true | false | true | true | true | true | false | false |

true | true | false | false | true | false | true | false | false |

true | false | true | true | false | true | false | true | true |

true | false | true | false | false | true | true | true | true |

true | false | false | true | false | true | true | false | true |

true | false | false | false | false | true | true | false | true |

false | true | true | true | false | true | false | false | false |

false | true | true | false | false | true | true | false | false |

false | true | false | true | false | true | true | false | false |

false | true | false | false | false | true | true | false | false |

false | false | true | true | false | true | false | false | true |

false | false | true | false | false | true | true | false | true |

false | false | false | true | false | true | true | false | true |

false | false | false | false | false | true | true | false | true |

It can be seen that when all premises are true then conclusion is also true. Thus argument is valid.

**(c)**

Warning Light is on, Pressure is too high, Relief valve is clogged.

true | true | true | true | false | true |

true | true | false | false | true | true |

true | false | true | false | false | false |

true | false | false | false | true | false |

false | true | true | false | false | false |

false | true | false | true | true | false |

false | false | true | true | false | true |

false | false | false | true | true | true |

It can be seen that in one row above, when all premises are true, then corr. conclusion is not true. Thus argument is not valid.

**Soln5**

**(a)**

is equivalent to

**(b)**

.

**Soln6**

**(a)**

.

**(b)**

.

**Soln7**

**(a)**

RHS:

Consider:

- In above equation, Q is present two times, and .
- The equation will be true when all of the terms and and are true.
- That means and should be true. Here is present in both terms, and . So for both terms to become true, and must be true.
- Thus we have from the two terms containing , that must be true, for these two terms to become true.
- That means we can safely remove the term from the equation as it will be true if next two terms are also true.

Thus we have:

= LHS.

**(b)**

.

**Soln8**

.

**Soln9**

Using Soln8,

.

**Soln10**

**(a)**

.

**(b)**

. This is equivalent to (a).

**(c)**

**(d)**

. This is equivalent to (a).

**(e)**

. This is equivalent to (c).