Proving Equivalence of p and (q\Rightarrowr)

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In summary, the OP is given two statements: p and (q ##\Rightarrow## r) and needs to prove their equivalence. They know that p ##\Rightarrow## (q ##\Rightarrow## r) is equivalent to (p ##\wedge## q) ##\Rightarrow## r, but are unsure how to rewrite the converse, (q ##\Rightarrow## r) ##\Rightarrow## p. The suggestion is to convert p ##\Rightarrow## q to (not (p ##\wedge## not q)) and use Boolean algebra to rework the expression, then convert back. However, there may be missing information as it is unclear how arbitrary statements could be proven equivalent. The suggestion to use a truth table
  • #1
jdm900712
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[itex]\Rightarrow[/itex]

Homework Statement


I'm given two statements, p and (q[itex]\Rightarrow[/itex]r) and I need to prove that the two statements are equivalent. So I need to show that p [itex]\Leftrightarrow[/itex](q[itex]\Rightarrow[/itex]r)

I know that p[itex]\Rightarrow[/itex](q[itex]\Rightarrow[/itex]r) [itex]\Leftrightarrow[/itex] (p[itex]\wedge[/itex]q)[itex]\Rightarrow[/itex]r
but I don't know how I should rewrite the converse:
(q[itex]\Rightarrow[/itex]r)[itex]\Rightarrow[/itex]p

Homework Equations





The Attempt at a Solution

 
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  • #2
What if you convert the p --> q to ( not (p and not q) ) and then use Boolean algebra to rework the expression and then convert back.
 
  • #3
jdm900712 said:
[itex]\Rightarrow[/itex]

Homework Statement


I'm given two statements, p and (q[itex]\Rightarrow[/itex]r) and I need to prove that the two statements are equivalent. So I need to show that p [itex]\Leftrightarrow[/itex](q[itex]\Rightarrow[/itex]r)

I know that p[itex]\Rightarrow[/itex](q[itex]\Rightarrow[/itex]r) [itex]\Leftrightarrow[/itex] (p[itex]\wedge[/itex]q)[itex]\Rightarrow[/itex]r
but I don't know how I should rewrite the converse:
(q[itex]\Rightarrow[/itex]r)[itex]\Rightarrow[/itex]p

Homework Equations





The Attempt at a Solution


I think there is some information that is missing here. I don't see how the arbitrary statements p and (q ##\Rightarrow## r) could be equivalent.

For example, let p, q and r be the following statements:
p: x = 2
q: y = 5
r: y2 = 25

Whether p is true or false has no bearing on the implication q ##\Rightarrow## r

Are p, q, and r specific statements that aren't given in the OP?
 
  • #4

1. What is the definition of "equivalence" in the context of p and (q⇒r)?

Equivalence refers to the logical relationship between two statements or propositions, where they have the same truth value in all possible cases. In the context of p and (q⇒r), it means that p is true if and only if the conditional statement q implies r is also true.

2. How do you prove equivalence between p and (q⇒r)?

To prove equivalence between p and (q⇒r), you can use a truth table or logical equivalences such as De Morgan's laws and the Law of Contrapositive. You can also use the method of direct proof, where you assume p is true and show that (q⇒r) is also true, and vice versa.

3. What are the implications of proving equivalence between p and (q⇒r)?

If p and (q⇒r) are proven to be equivalent, it means that p and (q⇒r) will always have the same truth value. This means that whenever p is true, (q⇒r) will also be true, and whenever p is false, (q⇒r) will also be false. This allows for simplification and transformation of logical statements in mathematical proofs and reasoning.

4. Can you give an example of proving equivalence between p and (q⇒r)?

For example, let p be the statement "It is raining", q be "I bring an umbrella", and r be "I stay dry". We can prove equivalence between p and (q⇒r) by showing that whenever p is true (it is raining), (q⇒r) is also true (if I bring an umbrella, I stay dry), and whenever p is false (it is not raining), (q⇒r) is also false (if I don't bring an umbrella, I may not stay dry).

5. Why is proving equivalence between p and (q⇒r) important in scientific research?

In scientific research, proving equivalence between p and (q⇒r) can help to simplify and clarify complex logical statements and arguments. It allows scientists to make logical deductions and draw conclusions based on the equivalence of two statements, which can aid in the development of theories and hypotheses. Additionally, it allows for consistency and accuracy in the communication of scientific ideas and findings.

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