Assignment 1 & 2

Since Assignment -I and II  is not properly visible on mobile devices, I am sharing these snapshots.

Rules of Inference

An argument (in propositional logic) is a sequence of propositions. All but the final proposition are called premises. The last proposition is the conclusion. The argument is valid iff the truth of all premises implies the conclusion is true.

In this discussion, we are using a notation where, hypotheses are written in a column, followed by a horizontal bar, followed by a line that begins with the therefore symbol with the conclusion.

Addition

P

----------------

∴P∨Q

Disjunctive Syllogism

P∨Q

¬P

----------------

∴Q

Conjunction

P

Q

----------------

∴P∧Q

Hypothetical Syllogism

P→Q

Q→R

----------------

∴ P→R

Simplification

P∧Q

----------------

∴P

Constructive Dilemma

(P→Q)∧(R→S)

P∨R

----------------

∴Q∨S

Modus Ponens

P→Q

P

----------------

∴Q

Destructive Dilemma

(P→Q)∧(R→S)

¬Q∨¬S

----------------

∴¬P∨¬R

Modus Tollens

P→Q

¬Q

----------------

∴¬P

Resolution

P∨Q

¬P∨R

----------------

∴ Q∨R

How to build arguments using the rules of inference

1. I t is not sunny this afternoon and it is colder than yesterday.

2. If we go swimming it is sunny.

3. If we do not go swimming then we will take a canoe trip.

4. If we take a canoe trip then we will be home by sunset.

5. We will be home by sunset

Ref: Discrete Mathematics and Its Applications (SIE) by Kenneth H Rosen

Discrete Mathematics Assignment - II

Discrete Mathematics	AY 2018-19
Assignment –II	Date of submission:  11/08/2018

Q1. Let R be a symmetric relation. Show that Rⁿ is symmetric for all positive integers n.

Q2. Let R be the relation on the set of all urls such that xRy iff the web page at x is the same as the web page at y. show that R is an equivalence relation.

Q3. Let R be reflexive relation on set A. Show that R Í  R²

Q4. Determine whether each of these functions is a bijection from R to R

i.                   f(x)= 2x+1

ii.                 f(x)= x²+1

iii.              f(x)= x³

Discrete Mathematics Assignment -I

Discrete Mathematics	AY 2018-19
Assignment –I	Date of submission:  11/08/2018

Q1. Show that for any two sets A & B

                                    P(A) È P(B) Í  P(AÈB)

Q2. Prove that

                                    (AÇ B) X (CÇ D) = (AXC) Ç (BXD)

Q3. Let A and B be subsets of universal set U.

Show that A ÍB if and only if B’Í A’

Q4. State the rules of inference.

Q5. Using rules of inference, show that conclusion follows from hypothesis.

            Hypothesis:

            P: if there is a gas in car then I will go to store.

            Q: if I go to the store, then I will get soda.

            R: there is gas in car.

            Conclusion:    

                        I will get soda.

Q6. Using rules of inference, show that conclusion follows from hypothesis.

            Hypothesis:

            P: Everyone in class has graphing calculator.

            Q: Everyone who has graphing calculator, understands trigonometric function.

Conclusion:

            Avyaan, who is in class, understands trigonometric function.