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.**

**N**ow 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).**