Question
Show that the following propositions are logically equivalent using logical transformations (not using truth tables) p→q≡~q→~p
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Answer to a math question Show that the following propositions are logically equivalent using logical transformations (not using truth tables) p→q≡~q→~p
Frederik
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To show that the propositions p→q and ~q→~p are logically equivalent, we can apply logical transformations.
Starting with p→q, we can use the definition of implication to rewrite it as ~p∨q.
Next, we can use De Morgan's law to further transform it as ~(p∧~q).
Now let's look at ~q→~p. Using the definition of implication, we can rewrite it as ~(~q)∨~p.
Applying double negation, ~(~q) becomes q, so ~q→~p can be written as q∨~p.
By comparing ~(p∧~q) with q∨~p, we can see that they are the negations of each other. Therefore, they are logically equivalent.
Answer: p→q≡~q→~p
Starting with p→q, we can use the definition of implication to rewrite it as ~p∨q.
Next, we can use De Morgan's law to further transform it as ~(p∧~q).
Now let's look at ~q→~p. Using the definition of implication, we can rewrite it as ~(~q)∨~p.
Applying double negation, ~(~q) becomes q, so ~q→~p can be written as q∨~p.
By comparing ~(p∧~q) with q∨~p, we can see that they are the negations of each other. Therefore, they are logically equivalent.
Answer: p→q≡~q→~p
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