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Let P (x) be the statement "x can speak Russian" and let


Q(x) be the statement "x knows the computer language


C++." Express each of these sentences in terms of P (x),


Q (x), quantifiers, and logical connectives. The domain


for quantifiers consists of all students at your school.



{x/x is an integer such as x2 =2}


How many rows appear in a truth table for each of these

compound propositions?

a) (q -+ -'p) v (-'P -+ -.q)

b) (p v -.t) /\ (p v -'s)

c) (p -+ r) V (-,s -+ -.t) v (-,u -+ v)

d) (p /\ r /\ s) V (q /\ t) V (r /\ -.t)


Formulate the symbolic expression in English Sentence using p:Today is Monday. q:It is raining. r:It is hot.

4. ¬→ (q ˅ r)

5. ¬→ r

6. q → ¬r

  1. Determine whether each of the following statements is a proposition or not. If it is, give its truth value.p: Mindanao is an island in the Philippines. 
  • q: Find a number which divides your age. 
  • r: My seatmate will get a perfect score in the Logic exam.
  • s: Welcome to the Philippines!




show that "p \\leftrightarrow q" and "(p \\land q) \\lor (\\neg p \\land \\neg q)" are logically equivalent.


find the truth value "1\\land 1 \\to 1 \\lor 1"

show that

"(p\\leftrightarrow q) (p\\land q) \\lor (\\neg p \\land \\neg q)"

are logically equivalent



The country’s capital city of Kathmandu is located in the center of Province 3. You decide to visit Nepal and want to drive through all provinces, trying to avoid as much as possible to visit one province twice. You arrive at Kathmandu Airport and take a rental car. The trip can begin. But wait – let’s plan things a little bit first. You open your map and your laptop.

As an abstract thinker, you may decide to first formalize the map as a graph and use it to plan the trip.


1)   Formalize the map of Nepal as a graph , where the vertices are the provinces and edges connect vertices that represent different provinces with a common border. For example, {1,3} is in E because Provinces 1 and 3 share a border.

2)   What degree does each vertex of G have?

3)   Determine which province is adjacent to the most other provinces.

4)   Is G planar? If yes, provide a drawing as a justification, that is, draw G so that edges meet only at vertices. If the graph is planar, how many faces does it have?



Draw a graph which has an Euler circuit but is not planar. Formalize the graph in the form 

Draw a graph which does not have an Euler path and is also not planar. Formalize the graph in the form 


Note: If you cannot draw the graph due to technical reasons, it is OK to just use formal notation and describe the graph textually.