QA 14

#$&*

course MTH 151

11:25 Am 3/4/2015

014. Truth Tables

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Question: `q001. There are 10 questions in this set.

If each of the propositions p and q can be either true or false, what combinations of truth values are possible for the two propositions (e.g., one possibility is that p is false and q is true; list the other possibilities)?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

TT TF FF FT

confidence rating #$&*:3

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Given Solution:

It is possible that p is true and q is true.

Another possibility is that p is true and q is false.

A third possibility is that p is false and q is true.

A fourth possibility is that p is false and q is false.

These possibilities can be listed as TT, TF, FT and FF, where it is understood that the first truth value is for p and the second for q.

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Self-critique (if necessary):

ok

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Self-critique Rating:3

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Question: `q002. For each of the for possibilities TT, TF, FT and FF, what is the truth value of the compound statement p ^ q ?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

TT is true , TF is false , FT is false , and FF is false.

confidence rating #$&*:3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:ok

p ^ q means 'p and q', which is only true if both p and q are true.

In the case TT, p is true and q is true so p ^ q is true.

In the case TF, p is true and q is false so p ^ q is false.

In the case FT, p is false and q is true so p ^ q is false.

In the case FF, p is false and q is false so p ^ q is false.

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Self-critique (if necessary):

ok

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Self-critique Rating:3

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Question: `q003. Write the results of the preceding problem in the form of a truth table.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

p q p^q

T T T

T F F

F T F

F F F

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

The truth table must have headings for p, q and p ^ q. It must include a line for each of the possible combinations of truth values for p and q. The table is as follows:

p q p ^ q

T T T

T F F

F T F

F F F.

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Self-critique (if necessary):

okay

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Self-critique Rating:3

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Question: `q004. For each of the possible combinations TT, TF, FT, FF, what is the truth value of the proposition p ^ ~q?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

T T is F

T F is true

F T is false

F F is false

confidence rating #$&*:3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

For TT we have p true, q true so ~q is false and p ^ ~q is false.

For TF we have p true, q false so ~q is true and p ^ ~q is true.

For FT we have p false, q true so ~q is false and p ^ ~q is false.

For FF we have p false, q false so ~q is true and p ^ ~q is false.

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Self-critique (if necessary):

ok

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Self-critique Rating:3

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Question: `q005. Give the results of the preceding question in the form of a truth table.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

p q ~q p^~q

T T F F

T F T T

F T F F

F F T F

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

The truth table will have to have headings for p, q, ~q and p ^ ~q. We therefore have the following:

p q ~q p^~q

T T F F

T F T T

F T F F

F F T F

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Self-critique (if necessary):

ok

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Self-critique Rating:3

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Question: `q006. Give the truth table for the proposition p U q, where U stands for disjunction.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

P Q P U Q

T T T

T F T

F T T

F F F

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

p U q means 'p or q' and is true whenever at least one of the statements p, q is true. Therefore p U q is true in the cases TT, TF, FT, all of which have at least one 'true', and false in the case FF. The truth table therefore reads

p q p U q

T T T

T F T

F T T

F F F

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Self-critique (if necessary):

ok

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Self-critique Rating:3

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Question: `q007. Reason out the truth values of the proposition ~(pU~q).

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

TT p is true and q is true, so ~q is false. p U ~q is true. ~(p U ~q) is false.

TF p is true and q is false, so ~q is true. p U ~q is true, (as is q). ~(p U ~q) is false.

FT p is false and q is true, so ~q is false. p U ~q is false, ~(p U ~q) is true.

FF p is false and q is false ~q is true p U ~q is true ~(p U ~q) is false.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

In the case TT p is true and q is true, so ~q is false. Thus p U ~q is true, since p is true. So ~(p U ~q) is false.

In the case TF p is true and q is false, so ~q is true. Thus p U ~q is true, since p is true (as is q). So ~(p U ~q) is false.

In the case FT p is false and q is true, so ~q is false. Thus p U ~q is false, since neither p nor ~q is true. So ~(p U ~q) is true.

In the case FF p is false and q is false, so ~q is true. Thus p U ~q is true, since ~q is true. So ~(p U ~q) is false.

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Self-critique (if necessary):ok

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Self-critique Rating:3

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Question: `q008. Construct a truth table for the proposition of the preceding question.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

p q ~q pU~q ~(pU~q)

T T F T F

T F T T F

F T F F T

F F T T F

confidence rating #$&*:3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

We need headings for p, q, ~q, p U ~q and ~(p U ~q). Our truth table therefore read as follows:

p q ~q pU~q ~(pU~q)

T T F T F

T F T T F

F T F F T

F F T T F

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Self-critique (if necessary):

ok

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Self-critique Rating:

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Question: `q009. Construct a truth table for the statement (p ^ ~q).

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

P Q U ~Q p U ~Q ( P ^ ~ Q)

T T F F T

@&

If P and Q are both true, then ~Q is false. A conjunction being true only when both statements are true, the conjunction is false in this case.

*@

T F T T F

@&

In the line above P is true and Q is false, so ~Q is true. This makes the conjunction true.

*@

F T F F T

@&

In the third line, above, P is false and Q is true. So both P and ~Q would be false.

A conjunction is true only when both statements are true.

*@

F F T F F

confidence rating #$&*:

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Question: `q010. Construct a truth table for the statement q U (p ^ ~q).

P Q U Q P U ~Q ( P ^ ~ Q )

T T T F F

T F F T T

F T T F F

F F F F F

@&

Your table would be valid for P U ~Q. It isn't valid for P ^ ~Q.

*@

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:2

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

2

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Self-critique Rating:

&#Your work looks good. See my notes. Let me know if you have any questions. &#

QA 14

#$&*

course MTH 151

11:28 3/4/23015

014. Truth Tables

*********************************************

Question: `q001. There are 10 questions in this set.

If each of the propositions p and q can be either true or false, what combinations of truth values are possible for the two propositions (e.g., one possibility is that p is false and q is true; list the other possibilities)?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

TT TF FF FT

confidence rating #$&*:3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

It is possible that p is true and q is true.

Another possibility is that p is true and q is false.

A third possibility is that p is false and q is true.

A fourth possibility is that p is false and q is false.

These possibilities can be listed as TT, TF, FT and FF, where it is understood that the first truth value is for p and the second for q.

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Self-critique (if necessary):

ok

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Self-critique Rating:3

*********************************************

Question: `q002. For each of the for possibilities TT, TF, FT and FF, what is the truth value of the compound statement p ^ q ?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

TT is true , TF is false , FT is false , and FF is false.

confidence rating #$&*:3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:ok

p ^ q means 'p and q', which is only true if both p and q are true.

In the case TT, p is true and q is true so p ^ q is true.

In the case TF, p is true and q is false so p ^ q is false.

In the case FT, p is false and q is true so p ^ q is false.

In the case FF, p is false and q is false so p ^ q is false.

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Self-critique (if necessary):

ok

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Self-critique Rating:3

*********************************************

Question: `q003. Write the results of the preceding problem in the form of a truth table.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

p q p^q

T T T

T F F

F T F

F F F

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

The truth table must have headings for p, q and p ^ q. It must include a line for each of the possible combinations of truth values for p and q. The table is as follows:

p q p ^ q

T T T

T F F

F T F

F F F.

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Self-critique (if necessary):

okay

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Self-critique Rating:3

*********************************************

Question: `q004. For each of the possible combinations TT, TF, FT, FF, what is the truth value of the proposition p ^ ~q?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

T T is F

T F is true

F T is false

F F is false

confidence rating #$&*:3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

For TT we have p true, q true so ~q is false and p ^ ~q is false.

For TF we have p true, q false so ~q is true and p ^ ~q is true.

For FT we have p false, q true so ~q is false and p ^ ~q is false.

For FF we have p false, q false so ~q is true and p ^ ~q is false.

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Self-critique (if necessary):

ok

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Self-critique Rating:3

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Question: `q005. Give the results of the preceding question in the form of a truth table.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

p q ~q p^~q

T T F F

T F T T

F T F F

F F T F

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

The truth table will have to have headings for p, q, ~q and p ^ ~q. We therefore have the following:

p q ~q p^~q

T T F F

T F T T

F T F F

F F T F

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Self-critique (if necessary):

ok

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Self-critique Rating:3

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Question: `q006. Give the truth table for the proposition p U q, where U stands for disjunction.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

P Q P U Q

T T T

T F T

F T T

F F F

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

p U q means 'p or q' and is true whenever at least one of the statements p, q is true. Therefore p U q is true in the cases TT, TF, FT, all of which have at least one 'true', and false in the case FF. The truth table therefore reads

p q p U q

T T T

T F T

F T T

F F F

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Self-critique (if necessary):

ok

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Self-critique Rating:3

*********************************************

Question: `q007. Reason out the truth values of the proposition ~(pU~q).

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

TT p is true and q is true, so ~q is false. p U ~q is true. ~(p U ~q) is false.

TF p is true and q is false, so ~q is true. p U ~q is true, (as is q). ~(p U ~q) is false.

FT p is false and q is true, so ~q is false. p U ~q is false, ~(p U ~q) is true.

FF p is false and q is false ~q is true p U ~q is true ~(p U ~q) is false.

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

In the case TT p is true and q is true, so ~q is false. Thus p U ~q is true, since p is true. So ~(p U ~q) is false.

In the case TF p is true and q is false, so ~q is true. Thus p U ~q is true, since p is true (as is q). So ~(p U ~q) is false.

In the case FT p is false and q is true, so ~q is false. Thus p U ~q is false, since neither p nor ~q is true. So ~(p U ~q) is true.

In the case FF p is false and q is false, so ~q is true. Thus p U ~q is true, since ~q is true. So ~(p U ~q) is false.

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Self-critique (if necessary):ok

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Self-critique Rating:3

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Question: `q008. Construct a truth table for the proposition of the preceding question.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

p q ~q pU~q ~(pU~q)

T T F T F

T F T T F

F T F F T

F F T T F

confidence rating #$&*:3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

We need headings for p, q, ~q, p U ~q and ~(p U ~q). Our truth table therefore read as follows:

p q ~q pU~q ~(pU~q)

T T F T F

T F T T F

F T F F T

F F T T F

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Self-critique (if necessary):

ok

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Self-critique Rating: