DETERMINE WHICH IF ANY,OF THE THREE STATEMENTS ARE EQUIVALENT. GIVE A REASON FOR YOUR CONCLUSION.SHOW COMPLETE WORK. 1) iF THE DOG LICKS HER NOSE, THEN THE DOG IS NOT HAPPY. 2) IF THE DOG LICKS HER NOSE,THEN THE DOG IS HAPPY. 3) iF THE DOG IS NOT HAPPY,THEN THE DOG DOES NOT LICK HER NOSE.
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To determine whether the given statements are equivalent, we need to evaluate the logical connections between them. Let's examine each statement and its logical implications one by one:
Statement 1: "If the dog licks her nose, then the dog is not happy."
This statement implies that if the dog performs the action of licking her nose, then it is not happy.
Statement 2: "If the dog licks her nose, then the dog is happy."
This statement implies that if the dog licks her nose, then it is happy.
Statement 3: "If the dog is not happy, then the dog does not lick her nose."
This statement implies that if the dog is not happy, then it does not perform the action of licking her nose.
Now, let's analyze the logical connections between the statements. To do this, we can consider each statement's contrapositive, which is logically equivalent:
Contrapositive of Statement 1: "If the dog is happy, then the dog does not lick her nose."
Contrapositive of Statement 2: "If the dog is not happy, then the dog does not lick her nose."
Contrapositive of Statement 3: "If the dog licks her nose, then the dog is happy."
By comparing the contrapositives, we can see that the negated versions of Statements 1 and 3 are equivalent to the original Statement 2. Therefore, Statements 1 and 3 are equivalent, while Statement 2 is not equivalent to the others.
In summary:
- Statements 1 and 3 are equivalent because their contrapositives match Statement 2.
- Statement 2 is not equivalent to Statements 1 and 3.
Please note that logical equivalence is determined by comparing the logical implications and connections between statements, rather than the truth value of the statements themselves.