# Discrete Math - HELP

• Nov 8th 2008, 05:55 AM
tukilala
Discrete Math - HELP
this is the question:

If you voted Obama then you have a proud Mama."

John did not vote Obama, and has a proud Mama.
Joe voted Obama, and has a proud Mama.
Sarah did not vote Obama, and does not have a proud Mama.
Decide if the above statement is true for each of these people.
Do the same for the negation, the converse, and the contrapositive.

how can i slove it??
• Nov 8th 2008, 08:59 AM
Soroban
Hello, tukilala!

I'll get you started on this one . . .

Quote:

If you voted Obama, then you have a proud Mama.

(a) John did not vote Obama and has a proud Mama.
(b) Joe voted Obama and has a proud Mama.
(c) Sarah did not vote Obama and does not have a proud Mama.

[1] Decide if the above statement is true for each of these people.

[2] Do the same for the negation, the converse, and the contrapositive.

$\text{We have: }\:\text{If }\underbrace{\text{you voted Obama}}_p\text{, then }\underbrace{\text{you have a proud Mama.}}_q$

So we have: . $p \to q\;\text{ which is equivalent to }\sim p \vee q$

(a) John did not vote Obama and has a proud Mama: . $\sim p \wedge q$ . . . True

(b) Joe voted Obama and has a proud Mama: . $p \wedge q$ . . . True

(c) Sarah did not vote Obama and does not have a proud Mama: . $\sim p\: \wedge \sim q$ . . . True

• Nov 8th 2008, 03:22 PM
poutsos.B
Quote:

Originally Posted by tukilala
this is the question:

If you voted Obama then you have a proud Mama."

John did not vote Obama, and has a proud Mama.
Joe voted Obama, and has a proud Mama.
Sarah did not vote Obama, and does not have a proud Mama.
Decide if the above statement is true for each of these people.
Do the same for the negation, the converse, and the contrapositive.

how can i slove it??

The statement :

...."if you voted Obama then you have a good Mama "................

is a conditional one .................................................. ............1

Let us put : you vote Obama=p .......and you have a good Mama=q

For the case where:

john did not vote Obama, and has a proud Mama,

p is false and q is true hence according to the definition of the conditional .........................(1) is true

For the next case :.........p is true and q is true hence:

......................(1) is true.

For the next case :............p is false and q is false hence:

.........................(1) is true.

Now the negation of (1) is ~(~pvq) <====> (by De Morgan) p^ ~q ,since p---->q is equivalent to ~pvq.............................................. ......................2

The new statement now is not a conditional anymore but a conjunct ,hence using the definition of a conjunct statement for:

The first case: .................p is false and q is false hence:

..............................(2) is false.

For the next case :..............p is true and q is false hence:

.....................................(2) is false.

For the next case:..............p is false and q is is true hence:

.....................................(2) is false.

Now the converse of (1) is the conditional, q------->p....................3

And:

For the first case :.........again q is true and p is false ,but now :

....................................(3) is false.

For the next case :............q is true and p is true hence:

......................................(3) is true.

For the next case:...............q is false and p is false hence:

.......................................(3) is true.

The contrapositive of (1) is ~q------>~p.................................4

But now since (1) and (4) are equivalent they have the same truth table and hence for all the above cases (4) has the same values as (1).