DISCRETE MATH: Determine if two statements are logically equivalent

In summary, the two statements \forall\,x\,(P(x)\,\longleftrightarrow\,Q(x)) and \forall\,x\,P(x)\,\longleftrightarrow\,\forall\,x\,Q(x) are not logically equivalent. This can be shown by setting P and Q to always be TRUE, which makes both statements equivalent, but when P and Q are always FALSE, the statements are no longer equivalent. It is important to properly write out statements using logical symbols to avoid confusion.
  • #1
VinnyCee
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Homework Statement



Determine whether [tex]\forall\,x\,(P(x)\,\longleftrightarrow\,Q(x))[/tex] and [tex]\forall\,x\,P(x)\,\longleftrightarrow\,\forall\,x\,Q(x)[/tex] are logically equivalent. Justify your answer.

Homework Equations



[tex]P\,\longleftrightarrow\,Q[/tex] is only TRUE when both P and Q are TRUE or FALSE.

The Attempt at a Solution



No, I don't think the two statements are logically equivalent, but I have trouble trying to "justify" my answer.

Set P and Q as always TRUE.

Both statements are equivalent, but if you set P and Q to always FALSE, then the statements are no longer equivalent.

Does this seem logical:rolleyes:
 
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  • #2
Not very. Why not just write things out properly? As in for all z in S then P(z)
is just the same as z in S implies P(z).
 

1. What is discrete math and how is it used in determining logical equivalence?

Discrete math is a branch of mathematics that deals with discrete objects and structures, rather than continuous ones. It is used in determining logical equivalence by providing a formal language and rules for representing and manipulating logical statements.

2. How do you determine if two statements are logically equivalent?

To determine if two statements are logically equivalent, you can use truth tables, logical equivalences, or proofs. These methods involve examining the logical structure and truth values of the statements to see if they are equivalent in all cases.

3. Can two statements be logically equivalent if they have different wording or symbols?

Yes, two statements can be logically equivalent even if they have different wording or symbols. As long as the logical structure and truth values of the statements are the same, they can be considered logically equivalent.

4. How important is it to determine logical equivalence in mathematics and computer science?

Determining logical equivalence is crucial in mathematics and computer science because it allows us to simplify and manipulate complex logical statements, making them easier to understand and work with. It is also important in ensuring the correctness and efficiency of algorithms and programs.

5. Can you provide an example of two logically equivalent statements?

Yes, for example, the statements "If it is raining, then the ground is wet" and "The ground is wet if it is raining" are logically equivalent. They both have the same logical structure and truth values in all cases.

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