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Thread: Implication and Equivalence basics

  1. #1
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    Implication and Equivalence basics

    hi,

    I'm embarking on a course on Coursera in Introduction to Mathematical Thinking. I seem to be going seriously wrong somewhere because none of my answers are coming back as being correctly answered.


    Q1 - . Build a truth table to prove the claim I made earlier that φ ⇔ ψ is true if φ and ψ are both true or both false, and φ ⇔ ψ is false if exactly one of φ, ψ is true and the other false. (To constitute a proof, your table should have columns that show how the entries for φ ⇔ ψ are derived, one operator at a time.)
    http://[Q1](https://i.imgur.com/WLi5xE8.jpg)

    Questions: If we can assume that F(alse) = 0, while T(rue) = 1 and equivalence is multiplying the two together then we'll get:
    T F => F
    T T => T
    F T => F
    F F => T

    but that means that 0 x 0 should equal 1, which it obviously doesn't yet I've heard consistently that F F => T -- can someone clarify where I'm going wrong or where I have misunderstood?
    Q2 -Build a truth table to show that (φ ⇒ ψ) ⇔ (φ ∨ ψ) is true for all truth values of φ and ψ. A statement whose truth values are all T is called a logical validity, or sometimes a tautology.
    http://[Q2](https://i.imgur.com/tnmvIPC.jpg)

    I'm not sure what went wrong?!
    Questions: Is there an easy way to ensure that I've included all the correct notations in the truth table?
    Q3 - Build a truth table to show that (φ 6⇒ ψ) ⇔ (φ ∧ ψ) is a tautology.
    http://[Q3](https://i.imgur.com/oJf7wkL.jpg)

    Again I'm not sure where I'm going wrong here!
    Jumping to 9) to stay on the same theme
    http://[Q4](https://i.imgur.com/QrwWU0k.jpg)

    I must be misunderstanding something here to keep getting these so wrong, any assistance would be much appreciated.
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  2. #2
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    Re: Implication and Equivalence basics

    Quote Originally Posted by MagicHat View Post
    Questions: If we can assume that F(alse) = 0, while T(rue) = 1 and equivalence is multiplying the two together then we'll get:
    T F => F
    T T => T
    F T => F
    F F => T

    but that means that 0 x 0 should equal 1, which it obviously doesn't yet I've heard consistently that F F => T -- can someone clarify where I'm going wrong or where I have misunderstood?
    You can do this for $\phi \wedge \psi$ (the AND operator). You cannot do it for any arbitrary operator. Treating T as 1 and F as 0 and multiplying is valid ONLY for the AND operator. The concept of AND being closely related to multiplication while OR is closely related to addition is nothing new. If you phrase the biconditional operator in terms of AND and OR, you can use multiplication and addition to prove your truth tables.

    Quote Originally Posted by MagicHat View Post
    Q2 -Build a truth table to show that (φ ⇒ ψ) ⇔ (φ ∨ ψ) is true for all truth values of φ and ψ. A statement whose truth values are all T is called a logical validity, or sometimes a tautology.
    http://[Q2](https://i.imgur.com/tnmvIPC.jpg)

    I'm not sure what went wrong?!
    Questions: Is there an easy way to ensure that I've included all the correct notations in the truth table?
    Q3 - Build a truth table to show that (φ 6⇒ ψ) ⇔ (φ ∧ ψ) is a tautology.
    http://[Q3](https://i.imgur.com/oJf7wkL.jpg)

    Again I'm not sure where I'm going wrong here!
    Jumping to 9) to stay on the same theme
    http://[Q4](https://i.imgur.com/QrwWU0k.jpg)

    I must be misunderstanding something here to keep getting these so wrong, any assistance would be much appreciated.
    I am not able to view your images.
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  3. #3
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    Re: Implication and Equivalence basics

    Quote Originally Posted by MagicHat View Post
    hi,

    I'm embarking on a course on Coursera in Introduction to Mathematical Thinking. I seem to be going seriously wrong somewhere because none of my answers are coming back as being correctly answered.


    Q1 - . Build a truth table to prove the claim I made earlier that φ ⇔ ψ is true if φ and ψ are both true or both false, and φ ⇔ ψ is false if exactly one of φ, ψ is true and the other false. (To constitute a proof, your table should have columns that show how the entries for φ ⇔ ψ are derived, one operator at a time.)
    http://[Q1](https://i.imgur.com/WLi5xE8.jpg)

    Questions: If we can assume that F(alse) = 0, while T(rue) = 1 and equivalence is multiplying the two together then we'll get:
    T F => F
    T T => T
    F T => F
    F F => T
    but that means that 0 x 0 should equal 1, which it obviously doesn't yet I've heard consistently that F F => T -- can someone clarify where I'm going wrong or where I have misunderstood?
    Q2 -Build a truth table to show that (φ ⇒ ψ) ⇔ (φ ∨ ψ) is true for all truth values of φ and ψ. A statement whose truth values are all T is called a logical validity, or sometimes a tautology.
    http://[Q2](https://i.imgur.com/tnmvIPC.jpg)

    I'm not sure what went wrong?!to
    Questions: Is there an easy way to ensure that I've included all the correct notations in the truth table?
    Q3 - Build a truth table to show that (φ 6⇒ ψ) ⇔ (φ ∧ ψ) is a tautology.
    http://[Q3](https://i.imgur.com/oJf7wkL.jpg)

    Again I'm not sure where I'm going wrong here!
    Jumping to 9) to stay on the same theme
    http://[Q4](https://i.imgur.com/QrwWU0k.jpg)

    I must be misunderstanding something here to keep getting these so wrong, any assistance would be much appreciated.
    All of your links are broken.
    Frankly without the links, I do not think we can possibly help you.
    What you posted is very confusing even for someone who has worked with this.
    It appears as though you are asking about an operator equivalent to the exclusive or ?
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    Re: Implication and Equivalence basics

    I'm not sure how to go about editing my own message and I can't find the answer in the FAQ's, I also can't seem to upload the attachments or insert a link - surely there are some instructions?
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    Re: Implication and Equivalence basics

    Quote Originally Posted by MagicHat View Post
    I'm not sure how to go about editing my own message and I can't find the answer in the FAQ's, I also can't seem to upload the attachments or insert a link - surely there are some instructions?
    I suggest that you simply repost. First click the $\boxed{\text{Go Advanced}}$ tab.
    Then scroll to the very bottom where that is an ATTACHMENTS BOX, inside of which is a $\boxed{\text{Manage Attachments} }$ tab.
    Select file and upload it, follow instructions.
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    Re: Implication and Equivalence basics

    Quote Originally Posted by Plato View Post
    I suggest that you simply repost. First click the $\boxed{\text{Go Advanced}}$ tab.
    Then scroll to the very bottom where that is an ATTACHMENTS BOX, inside of which is a $\boxed{\text{Manage Attachments} }$ tab.
    Select file and upload it, follow instructions.
    Yes, I already tried that but the uploader won't allow me to upload - it just stops without offering a reason why. I think the file is too big.
    Last edited by MagicHat; Oct 24th 2017 at 07:43 AM.
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  7. #7
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    Last edited by MagicHat; Oct 24th 2017 at 07:42 AM.
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  8. #8
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    Re: Implication and Equivalence basics

    Quote Originally Posted by MagicHat View Post
    Yes, I already tried that but the uploader won't allow me to upload - it just stops without offering a reason why.
    add a file, choose a file, open, upload,

    Are you dealing with the stroke function?
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  9. #9
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    Re: Implication and Equivalence basics

    Quote Originally Posted by Plato View Post
    add a file, choose a file, open, upload,

    Are you dealing with the stroke function?
    I'm familiar with doing that - however as I said it's refusing to upload - it doesn't state why - having looked through the upload specs it seems that the file may be too large - I've compressed it - its still too large. I've tried to compress it again - it refuses - I don't know what else to do.

    I've never heard of the stroke function - the course I'm doing is looking at equivalence / implication.
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  10. #10
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    Re: Implication and Equivalence basics

    Quote Originally Posted by MagicHat View Post
    Implication and Equivalence basics-untitled.gif
    That is just the negation of the exclusive or.

    $\neg (\underline \vee ) \equiv \begin{array}{*{20}{c}} T&T&\| & T \\ T&F&\|& F \\ F&T&\| & F \\ F&F&\| & T \end{array}$


    You can change all this by learning to use LaTeX.
    Last edited by Plato; Oct 24th 2017 at 08:17 AM.
    Thanks from topsquark
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  11. #11
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    Re: Implication and Equivalence basics

    Quote Originally Posted by Plato View Post
    Click image for larger version. 

Name:	Untitled.gif 
Views:	3 
Size:	65.2 KB 
ID:	38228
    That is just the negation of the exclusive or.

    $\neg (\underline \vee ) \equiv \begin{array}{*{20}{c}} T&T&\| & T \\ T&F&\|& F \\ F&T&\| & F \\ F&F&\| & T \end{array}$


    You can change all this by learning to use LaTeX.
    That's some undertaking to do, just to present a problem on this forum.
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    Re: Implication and Equivalence basics

    Quote Originally Posted by MagicHat View Post
    That's some undertaking to do, just to present a problem on this forum.
    Naw, the basics are easy. I agree with Plato, though, you need to do something. I don't know what's up with your images. I can see them, but apparently not everyone can.

    -Dan
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    Re: Implication and Equivalence basics

    Thanks for the reply.

    So I think I cracked some of it, however I'm still struggling with certain aspects of Question 2

    https://i.imgur.com/wWn1jry.jpg

    This is my working. I've checked and double checked my arguments and I cannot get them to agree that they are equivalent.

    Furthermore, in part b of Q8 again the arguments don't agree with what I know to be true.

    https://i.imgur.com/Zv3moAR.jpg

    Similarly in Part 9

    https://i.imgur.com/qlvK7eR.jpg

    I can't work out where I am going wrong despite checking it multiple times ;(( I also don't know a great deal of maths language so for me to understand hence I've not put the issue into more mathematical sounding language.

    Cheers for any help.
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    Re: Implication and Equivalence basics

    I'll work on trying to find a YouTube video on the basics so that I can upload more efficiently.
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    Re: Implication and Equivalence basics

    Quote Originally Posted by MagicHat View Post
    Thanks for the reply.
    So I think I cracked some of it, however I'm still struggling with certain aspects of Question 2
    https://i.imgur.com/wWn1jry.jpg
    This is without a doubt one of the three most important identities in logic.
    $A \Rightarrow \;B \Leftrightarrow \neg A \vee B$

    $\begin{array}{*{20}{c}} A&B&{\neg A}&|&{A \Rightarrow B}& \Leftrightarrow &{\neg A \vee B} \\ \hline T&T&F&|&T&|&T \\ T&F&F&|&F&|&F \\ F&T&T&|&T&|&T \\ F&F&T&|&T&|&T \end{array}$

    Tell us exactly what you do not understand. Do in in detail.
    Thanks from topsquark
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