# Converse, Contrapositive and Negation for multiple Quantifiers

• MHB
• Sandra Tan
In summary: Thus, the symbolic form of the negation would be:$\forall p B(p) \wedge \neg(\exists j Q(J))$.In summary, the symbolic forms for the three parts are:Converse: (∃jQ(j)) → (∀pB(p))Contrapositive: ¬(∃jQ(j)) → ¬(∀pB(p))Negation: $\forall p B(p) \wedge \neg(\exists j Q(J))$.
Sandra Tan
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!

Sandra Tan said:
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 ($\wedge$).

## 1. What is the difference between converse, contrapositive and negation for multiple quantifiers?

The converse of a statement is formed by switching the subject and predicate of the original statement. The contrapositive is formed by both switching the subject and predicate and negating both. Negation is simply the denial or opposite of the original statement. When dealing with multiple quantifiers, these operations are applied to all of the quantifiers in the statement.

## 2. How do you symbolically represent converse, contrapositive and negation for multiple quantifiers?

The converse of a statement is represented by switching the quantifiers and the predicates. The contrapositive is represented by switching the quantifiers and predicates, and adding a negation to each. Negation is represented by placing a negation symbol in front of the entire statement. It is important to note that when dealing with multiple quantifiers, the order of the quantifiers matters and must be preserved in the symbolic representation.

## 3. What is the purpose of using converse, contrapositive and negation for multiple quantifiers?

Using these operations can help to prove or disprove the validity of a statement. By applying these operations, you can see if the original statement is equivalent to its converse, contrapositive, or negation. If they are all true, then the original statement is logically valid.

## 4. Can you give an example of how converse, contrapositive and negation for multiple quantifiers are used?

Sure, let's take the statement "All cats are mammals." The converse of this statement would be "All mammals are cats." The contrapositive would be "All non-mammals are non-cats." The negation would be "Not all cats are mammals." By applying these operations, we can see that the original statement is equivalent to its contrapositive, and the converse and negation are not equivalent.

## 5. What are some common mistakes when using converse, contrapositive and negation for multiple quantifiers?

One common mistake is forgetting to switch the quantifiers in the symbolic representation. Another mistake is incorrectly negating the predicates. It's important to carefully apply the operations to all quantifiers in the statement. Another mistake is assuming that the converse or contrapositive are always true, when in fact a statement and its converse or contrapositive can both be false.

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