# Converse, Contrapositive and Negation for multiple Quantifiers

#### Sandra Tan

##### New member
If every printer is busy then there is a job in the queue.

where B(p) = Printer p is busy and Q(j) = Print job j is queued.

When it's translated to symbol, we'll have (∀pB(p)) → (∃jQ(j)).

I'm trying to translate this statement to both English and symbol forms for Converse, Contrapositive and Negation.

Following is what i have got so far:
Converse
in words: If there is a job in the queue, then every printer is busy.
in symbol: (∃jQ(j)) → (∀pB(p))

Contrapositive
in words: If there is no job in the queue, then not every printer is busy.
in symbol: ¬(∃jQ(j)) → ¬(∀pB(p))

Negation
in words: Every printer is busy and there is no job in the queue.
in symbol: (not sure)

It's the symbol part that i'm not sure if they are correct or not. Any advice would be appreciated!

#### castor28

##### Well-known member
MHB Math Scholar
If every printer is busy then there is a job in the queue.

where B(p) = Printer p is busy and Q(j) = Print job j is queued.

When it's translated to symbol, we'll have (∀pB(p)) → (∃jQ(j)).

I'm trying to translate this statement to both English and symbol forms for Converse, Contrapositive and Negation.

Following is what i have got so far:
Converse
in words: If there is a job in the queue, then every printer is busy.
in symbol: (∃jQ(j)) → (∀pB(p))

Contrapositive
in words: If there is no job in the queue, then not every printer is busy.
in symbol: ¬(∃jQ(j)) → ¬(∀pB(p))

Negation
in words: Every printer is busy and there is no job in the queue.
in symbol: (not sure)

It's the symbol part that i'm not sure if they are correct or not. Any advice would be appreciated!
Hi Sandra,

The first two propositions are correct.

For the third one, the natural language is correct too. For the symbolic form, note that you already have (from the first two parts) the expression of the two parts of the statement:

Every printer is busy: $\forall p B(p)$
There is no job in the queue: $\neg(\exists j Q(J))$

and all you have to do is to connect these two proposition with AND ($\wedge$).