Discrete Mathematics Answers

A box contains 6 red, 8 green,10 black 11 yellow and 12 white balls. What is the minimum number of

balls we have select from box to guarantee that 9 balls are of the same colours.

State the Pigeonhole Principle. Prove that if six integers are selected from the set

[3,4,5,6,7,8,9,10,11,12] there must be two integer whose sum is fifteen.

A computer randomly chooses a binary string of length 6. What is the probability that exactly two characters in the string turn out to be 1s?

State the Pigeonhole Principle. Prove that if six integers are selected from the set

[3,4,5,6,7,8,9,10,11,12] there must be two integer whose sum is fifteen.

6. (a) Draw the logic circuit for the following expression: AB + A(B+C)

(b) Simplify the expression in 6(a) by using the rules of Boolean algebra provided in (5). (c) Draw the simplified logic gate circuit derived in (b)

Use rules of inference to show that the hypotheses ”If it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on,”, ”If the sailing race is held, then the trophy will be awarded,” and ”The trophy was not awarded” imply the conclusion ”It rained.”

Let A and B be subsets of a universal set U. Show that A ⊆ B if and only if B ⊆ A.

29. Prove or disprove that if m and n are integers such that mn = 1 then either m = 1 and n = 1 , or else m = - 1 and n = - 1 .

I. Create a truth table for all of the 11 Logical Equivalence.

II. Find if the following are logically equivalent

   1.(~p v q) ^ (~q) <=> ~(p v q)

   2.(p ^ ~q) v (~p v q) <=> T

Draw Hasse diagram repressing the partial order on {(a,b) : a\b } on {1,2,3,4,6,8,12}