Math  /  Algebra

Question1. Let tt : Green Day is on tour. rr : Green Day is recording a new CDC D.
Nrite in symbolic form: 3 points each a)Green Day is not on tour, but Green Day is recording a new CD. b) Green Day is not recording a new CD and Green Day is on tour.
2. Let p: Maria will go to the circus. q: Maria will go to the zoo. Write in symbolic form: a) Maria will go to the zoo or Maria will not go to the circus. b) Maria will not go to the circus or Maria will not go to the zoo.
3. Let pp : The portrait is a pastel. qq : The portrait is by Beth Anderson.

Write the following statements symbolically. a) If the portrait is a pastel, then the portrait is by Beth Anderson. b) If the portrait is by Beth Anderson, then the portrait is not a pastel.
Write Equivalent statements. a) If today is Thanksgiving, then tomorrow is Friday. b) I do not pay taxes and I vote. c) A person had cake or they had ice cream.

Studdy Solution

STEP 1

1. We are given propositions and need to translate English statements into symbolic logic.
2. We are using standard logical symbols: ¬\neg for "not", \land for "and", \lor for "or", and \rightarrow for "if...then".

STEP 2

1. Translate the given statements into symbolic form.
2. Write equivalent statements for given logical propositions.

STEP 3

Translate the statement "Green Day is not on tour, but Green Day is recording a new CD" into symbolic form using the given propositions. - The statement "Green Day is not on tour" translates to ¬t\neg t. - The statement "Green Day is recording a new CD" translates to rr. - The word "but" functions as "and" in logic, so we use \land.
The symbolic form is: ¬tr\neg t \land r.

STEP 4

Translate the statement "Green Day is not recording a new CD and Green Day is on tour" into symbolic form. - The statement "Green Day is not recording a new CD" translates to ¬r\neg r. - The statement "Green Day is on tour" translates to tt. - The word "and" is represented by \land.
The symbolic form is: ¬rt\neg r \land t.

STEP 5

Translate the statement "Maria will go to the zoo or Maria will not go to the circus" into symbolic form. - The statement "Maria will go to the zoo" translates to qq. - The statement "Maria will not go to the circus" translates to ¬p\neg p. - The word "or" is represented by \lor.
The symbolic form is: q¬pq \lor \neg p.

STEP 6

Translate the statement "Maria will not go to the circus or Maria will not go to the zoo" into symbolic form. - The statement "Maria will not go to the circus" translates to ¬p\neg p. - The statement "Maria will not go to the zoo" translates to ¬q\neg q. - The word "or" is represented by \lor.
The symbolic form is: ¬p¬q\neg p \lor \neg q.

STEP 7

Translate the statement "If the portrait is a pastel, then the portrait is by Beth Anderson" into symbolic form. - The statement "The portrait is a pastel" translates to pp. - The statement "The portrait is by Beth Anderson" translates to qq. - The phrase "If...then" is represented by \rightarrow.
The symbolic form is: pqp \rightarrow q.

STEP 8

Translate the statement "If the portrait is by Beth Anderson, then the portrait is not a pastel" into symbolic form. - The statement "The portrait is by Beth Anderson" translates to qq. - The statement "The portrait is not a pastel" translates to ¬p\neg p. - The phrase "If...then" is represented by \rightarrow.
The symbolic form is: q¬pq \rightarrow \neg p.

STEP 9

Write an equivalent statement for "If today is Thanksgiving, then tomorrow is Friday." - The equivalent contrapositive statement is "If tomorrow is not Friday, then today is not Thanksgiving."

STEP 10

Write an equivalent statement for "I do not pay taxes and I vote." - The equivalent statement using De Morgan's Laws is "It is not the case that I pay taxes or I do not vote."

STEP 11

Write an equivalent statement for "A person had cake or they had ice cream." - The equivalent statement using De Morgan's Laws is "It is not the case that a person did not have cake and they did not have ice cream."

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