Discrete Mathematics Answers

a)​Establish that: is a tautology.

​

​b)​Translate in two ways each of these statements into logical expressions ​using predicates, quantifiers, and logical connectives. First, let the ​domain consists of the students in your class and second, let it consist ​of all people.

​i) Everyone in your class has a mobile phone.

​ii) Somebody in your class has seen a foreign movie.

​iii) There is a person in your class who cannot swim.

​iv) All students in your class can solve quadratic equations.

​v) Some students in your class does not want to be rich.

​c)​Prove that if m + n and n + p are even integers, where m, n, and

​p are integers, then m + p is even. What kind of proof did you use?

​d)​Prove, by induction that: 2 divides whenever n is a positive

​integer.

Let  and  be the propositions:

: “Ahmad comes to the party”,

: “Salim comes to the party”,

: “Khalid comes to the party”,

: “Bader comes to the party”.

Write the following propositions using  and logical connectives:

a)  “Nether Ahmad nor Salim come to the party”.

b) “If Bader comes to the party, then Salim and Khalid come too”.

c)  “Khalid comes to the party only if Ahmad and Salim do not come”.

d) “Bader comes to the party if and only if Khalid comes and Ahmad doesn’t come”.

e)  “If Bader comes to the party, then, if Khalid doesn’t come then Ahmad comes”.

f)   “Khalid comes to the party provided that Bader doesn’t come, but, if Bader comes, then Salim doesn’t come”.

g)  “A necessary condition for Ahmad coming to the party is that, if Salim and Khalid aren’t coming, Bader comes”.

h) “Ahmad, Salim and Khalid come to the party if and only if Bader doesn’t come, but, if neither Ahmad nor Salim come, then Bader comes only if Khalid comes”.

⦁ Find all such that .

⦁ If is a zero-one matrix, find .

⦁ Find the inverse of the encrypting function

,

and use it to decrypt the message “ZRQUBFNB”.

An urn contains 5 red balls and 6 blue balls. A ball is drawn randomly, its colour is noted, and then the ball is replaced in the urn. If this process is repeated four times: 

⦁ What is the probability of getting two red balls and 2 blue balls?

⦁ What is the probability that all 4 balls are of the same colour?

⦁ What is the probability that the first red ball is drawn in the 4th trial?