Answer to Question #217539 in Discrete Mathematics for lio

Question

Write Inverse, Converse and Contrapositive, also apply implication law on the following statements:                                                                                                            

a)If it snows today. I will ski tomorrow.b)I come to class whenever there is going to be a quiz.c)I go to the beach whenever it is a sunny summer day.d) When I stay late, it is necessary that I sleep until noon.

 

Answer 1

$\text{if implication: } p\to q$

$\text{then }$

$\text{inverse: }\neg p\to \neg q$

$\text{converse: }q\to p$

$\text{contrapositive:}\neg q\to \neg p$

$\text{a)If it snows today. I will ski tomorrow.}$

$p:\text{ it snows today. }$

$q:\text{I will ski tomorrow.}$

$\neg p:\text{ it's not snowing today. }$

$\neg q:\text{ I won't ski tomorrow. }$

$\text{inverse: }\neg p\to \neg q$

$\text{If it's not snowing today. I won't ski tomorrow. }$

$\text{converse: }q\to p$

$\text{if I will ski tomorrow.Then it snows today.}$

$\text{contrapositive:}\neg q\to \neg p$$\text{ if I won't ski tomorrow.Then it's not snowing today }$

$\text{b)I come to class whenever there is going to be a quiz.}$

$\text{If there is a quiz.I come to class.}$

$p: \text{There is a quiz}$

$q:\text{I come to class.}$

$\neg p:\text{The quiz is not going}$

$\neg q:\text{I don't come to class.}$

$\text{inverse: }\neg p\to \neg q$

$\text{If the quiz is not going.I don't come to class.}$

$\text{converse: }q\to p$

$\text{If I come to class.The quiz is going}$

$\text{contrapositive:}\neg q\to \neg p$

$\text{If i don't come to class.The quiz is not going.}$

$\text{c)I go to the beach whenever it is a sunny summer day.}$

$\text{If it is a sunny summer day.I go to the beach.}$

$p: \text{it is a sunny summer day.}$

$q: \text{I go to the beach.}$

$\neg p :\text{it's not a sunny summer day now.}$

$\neg q :\text{I don't go to the beach.}$

$\text{inverse: }\neg p\to \neg q$

$\text{If it's not a sunny summer day now.I don't go to the beach.}$

$\text{converse: }q\to p$

$\text{If i go to the beach. It is a sunny summer day.}$

$\text{contrapositive:}\neg q\to \neg p$

$\text{If i don't go to the beach.It's not a sunny summer day now.}$

$\text{d) When I stay late, it is necessary that I sleep until noon.}$

$p:\text{I stay late.}$

$q:\text{ I sleep until noon.}$

$\neg p:\text{I don't stay late}$

$\neg q:\text{I don't sleep until noon.}$

$\text{inverse: }\neg p\to \neg q$

$\text{If i don't stay late.I don't sleep until noon.}$

$\text{converse: }q\to p$

$\text{ If i sleep until noon.I stay late.}$

$\text{contrapositive:}\neg q\to \neg p$

$\text{If i don't sleep until noon.I don't stay late}$