Discrete Mathematics Answers

Write down all derangements of the set \left\{ a,b,c,d \right\} and show that the number of derangements is the same as predicted by the recurrence D(n) = (n - 1)(D(n - 2) + D(n - 1)) with initial values D(1) = 0 and D(2) = 1. Hint: a derangement is a permutation of an ordered set where no element is in the same place as before. Example: \left\{ b,a,d,c \right\} is a derangement of \left\{ a,b,c,d \right\} because all of the letters positions have changed.

How can I use a venn diagram to represent (1,2,3,4,5) and (1,3,5)

What is the cardinality of each of these sets?

a. {a}

b. {{a}}

c. {a,{a}}  

d. {a,{a},{a,{a}}}  

e. {∅,{∅},{∅,{∅}}} 

“If compound X is boiling, then its temperature must be at least 150◦C.” Assuming that this

statement is true, which of the following must also be true?

a. If the temperature of compound X is at least 150◦C, then compound X is boiling.

b. If the temperature of compound X is less than 150◦C, then compound X is not boiling.

c. Compound X will boil only if its temperature is at least 150◦C.

d. If compound X is not boiling, then its temperature is less than 150◦C.

e. A necessary condition for compound X to boil is that its temperature be at least 150◦C.

f. A sufficient condition for compound X to boil is that its temperature be at least 150◦C.

How many assignments of truth values to p; q; r and w are there for which

((p → q) → r) → w is true? Guess a formula in terms of the number of variables.

Find the counter example, if possible to these universally quantified statements,where the domain for all variables consists of all integers.

Ax (x > 0 V x < 0)

Suppose that P(×) is x² > 1.What is the truth value of Ax P(x) where the universe of discourse consist of all integers?

1. Use the truth tables to verify these equivalence

If A = {1, 2, 3} and B = {4, 5, 6}, state which of the following is a relation from A to B.

(a) R₁ = {(1, 4); (2, 5); (6, 3)}

(b) R₂ = {(2, 5); (3, 6)}

(c) R₃ = {(6, 3); (5, 2); (4, 1)}

(d) R₄ = {(1, 5); (1, 6); (2, 4); (2, 6), (3, 4), (3, 5)}

For the following relation, write an equation that describes the connection between x (the first component in an ordered pair) and y (the second component in an ordered pair).

a. {(1, 1), (1, -1), (4, 2), (4, -2), (9, 3), (9, -3)}

b. {(1, 4), (2, 3), (3, 2), (4, 1), (-2, -3), (6, -1), (7, -2)}