Logical equivalencies involving ifs and nors

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In summary, The conversation discusses finding a compound proposition that is logically equivalent to p → q using only the logical operator ↓. The suggestion given is to use the NOT function obtained by p NOR p, which breaks the symmetry. Another suggested equivalency is ¬(¬p ↓ p) and it is found in a previous exercise. Finally, the solution is given as ((p ↓ p) ↓ q) ↓ ((p ↓ p) ↓ q) by using the equivalency ¬p ≡ p ↓p.
  • #1
nicnicman
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Can anyone solve this step by step, so I can see how it's done? I've been at for a while now and can't seem to get it. Here's the problem:

Find a compound proposition logically equivalent to p → q using only the logical operator ↓.

Thanks for the help. I'm really trying to get this stuff, but it's not coming easy.
 
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  • #2
The trick is to obtain a NOT function by p NOR p. That breaks the symmetry.
 
  • #3
Thanks for the suggestion, I finally got it.

If anyone's interested:

p → q
≡ ¬p ∨ p this is one of the common equivalencies given in my book by Deitel
≡ ¬(¬p ↓ p) this equivalency was found in a previous exercise
≡ ¬((p ↓ p) ↓ q) by ¬p ≡ p ↓p
≡ ((p ↓ p) ↓ q) ↓ ((p ↓ p) ↓ q) by ¬p ≡ p ↓p
 

Related to Logical equivalencies involving ifs and nors

1. What is a logical equivalence involving an "if" statement?

A logical equivalence involving an "if" statement is a statement that is equivalent to another statement if the "if" statement is true. For example, "If it is raining, then I will bring an umbrella" is logically equivalent to "I will bring an umbrella if it is raining." Both statements mean the same thing and are interchangeable.

2. What is a logical equivalence involving a "nor" statement?

A logical equivalence involving a "nor" statement is a statement that is equivalent to another statement if the "nor" statement is true. For example, "I will not go to the store nor will I buy groceries" is logically equivalent to "If I go to the store, then I will not buy groceries." Both statements mean the same thing and are interchangeable.

3. How are logical equivalencies involving "if" and "nor" statements used in programming?

In programming, logical equivalencies involving "if" and "nor" statements are often used to simplify complex conditional statements. By identifying logical equivalencies, programmers can write more concise and efficient code.

4. Can logical equivalencies involving "if" and "nor" statements be applied to other logical operators?

Yes, logical equivalencies can be applied to other logical operators, such as "or" and "and." For example, "If it is not sunny or I am not wearing sunscreen" is logically equivalent to "If it is not sunny, then I am wearing sunscreen." This principle can be applied to any combination of logical operators.

5. How can I determine if two statements are logically equivalent?

To determine if two statements are logically equivalent, you can use truth tables or logical equivalency laws. Truth tables involve listing all possible combinations of truth values for the statements and comparing the resulting truth values. Logical equivalency laws, such as De Morgan's laws and distribution laws, can also help identify equivalencies between statements.

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