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MAT1830 - Discrete Mathematics for Computer Science

Assignment #1

(1) Give a truth table for ( ¬a  → (b V ¬c)) ↔ a.   State whether the sentence is a tautology, a contradiction or neither and briefly explain why.                                     [4]

(2) Use laws of logic to show that ¬ (p → ¬q) V (r → q) V q is logically equivalent to p V q . Explain       each step fully.           [4]

(3) Consider the statement “If xy is even, then at least one of x and y is even.” In English (not using symbols), write down the statement’s contrapositive and then write down the statement’s negation. You may assume that x and y in the statement are integers.                     [2]

(4) On a certain day Burkard and Maryam both make purchases from the Nerdtshirts online store. Members of the Nerdtshirts loyalty program get a discount and free shipping on all purchases. Any purchases of over $99 get free shipping. On the 14th of March, all purchases get a discount.

These are the only situations in which Nerdtshirts offers discounts or free shipping. Let p be the proposition that Burkard is a member of the loyalty program.

Let q be the proposition that Maryam is a member of the loyalty program.

Let r be the proposition that Burkard’s purchase was over $99.

Let s be the proposition that Maryam’s purchase was over $99.

Let t be the proposition that the day is the 14th of March.

Write down propositions (using just p,q,r,s,t, brackets and logical connectives) corresponding to the following statements.

(i) Maryam does not get free shipping.      [2]

(ii) Burkard is not a member of the loyalty program but he gets a discount.                               [2]

(iii) Exactly one of Burkard or Maryam gets a discount.                                                               [2]

(iv) Neither Burkard nor Maryam spend over $99 but at least one of them gets free shipping.     [2]

(v) Burkard gets a discount but not free shipping while Maryam gets both a discount and free shipping.             [2]

[No explanation required but no partial marks for incorrect answers.]