Question 1

$\text { Prove that all the solutions to the equation } x ^ { 2 } = x + 1 \text { are irrational. }$

Accepted Answer

This equation is equivalent to (and therefore has the same solutions as) $x ^ { 2 } - x - 1 = 0$. By the quadratic formula, the solutions are exactly $( 1 \pm \sqrt { 5 } ) / 2$. If either of these were a rational number $r$, then we would have $\sqrt { 5 } = \pm ( 2 r - 1 )$. Since the rational numbers are closed under the arithmetic operations, this would tell us that $\sqrt { 5 }$ was rational, which we know from this chapter it is not.

Question 2

What is the truth value of (p ∨ q) → (p ∧ q) when both p and q are false?

Accepted Answer

When p and q are both false, so are (p V q) and (p ∧ q) . Hence (p V q) → (p∧q) is true.

Question 3

Prove that there are no solutions in positive integers to the equation x4 + y4 = 100.

Accepted Answer

If x⁴+y⁴=100, then both x and y must be less than 4 , since 4⁴=256. Therefore the only possible values for x and y are 1,2 , and 3 , and the fourth powers of these are 1,16 , and 81 . Since none of 1+1,1+16, 1+81,16+16,16+81, and 81+81 is 100 , there can be no solution

Question 4

Write the converse and contrapositive of the statement "If it is sunny, then I will go swimming."

Accepted Answer

The converse of the statement

Question 5

Prove each of the following statements. (a) The sum of two even integers is always even. (b) The sum of an even integer and an odd integer is always odd.

Accepted Answer

(a) Suppose that m and n are even intege

Question 6

Suppose that Q(x) is the statement "x + 1 = 2x." What are the truth values of ∀x Q(x) and ∃x Q(x)?

Accepted Answer

Since @#LAT-DLM& is true if an

Question 7

A stamp collector wants to include in her collection exactly one stamp from each country of Africa. If $I$ (s) means that she has stamp $s$ in her collection, $F ( s , c )$ means that stamp $s$ was issued by country $c$, the domain for $s$ is all stamps, and the domain for $c$ is all countries of Africa, express the statement that her collection satisfies her requirement. Do not use the $\exists$ ! symbol.

Accepted Answer

The answer of A stamp collector wants to include in...

Question 8

$\text { Let } \boldsymbol { A } = \left[ \begin{array} { l l l } 
1 & 2 & 3 \
0 & 1 & 4
\end{array} \right] \text { and } \boldsymbol { B } = \left[ \begin{array} { l l } 
1 & 2 \
0 & 1 \
2 & 3
\end{array} \right] \text {. Find } \boldsymbol { A } \boldsymbol { B } \text { and } \boldsymbol { B } \boldsymbol { A } \text {. Are they equal? }$

Accepted Answer

We have @#LAT-DLM& and @#LAT-DLM&. Obvio

Question 9

(a) Prove or disprove that a $6 \times 6$ checkerboard with four squares removed can be covered with straight triominoes.
(b) Prove or disprove that an $8 \times 8$ checkerboard with four squares removed can be covered with straight triominoes.

Accepted Answer

(a) The @#LAT-DLM& board with four squares removed

Question 10

Let $f ( n ) = 2 n + 1$. Is $f$ a one-to-one function from the set of integers to the set of integers? Is $f$ an onto function from the set of integers to the set of integers? Explain the reasons behind your answers.

Accepted Answer

If @#LAT-DLM&, then @#LAT-DLM&. It follows that @#LAT-DLM&.

Question 11

$$\text { Prove or disprove that if } A , B \text {, and } C \text { are sets then } A - ( B \cap C ) = ( A - B ) \cap ( A - C ) \text {. }$$

Accepted Answer

This is false. For a

Question 12

$$\text { Find the values of } \sum _ { j = 1 } ^ { 100 } 2 \text { and } \sum _ { j = 1 } ^ { 100 } ( - 1 ) ^ { j } \text {. }$$

Accepted Answer

The answer of \[\text { Find the values of }...

Question 13

Let $\boldsymbol { A } = \left[ \begin{array} { l l l } 1 & 0 & 1 \ 0 & 1 & 1 \ 1 & 1 & 0 \end{array} \right]$
and $\boldsymbol { B } = \left[ \begin{array} { l l l } 0 & 1 & 0 \ 0 & 1 & 1 \ 1 & 0 & 0 \end{array} \right]$. Find the join, meet, and Boolean product of these two zero-one matrices.

Accepted Answer

The join of @#LAT-DLM& and @#LAT-DLM& is

Question 14

Suppose that $f$ is the function from the set $\{ a , b , c , d \}$ to itself with $f ( a ) = d , f ( b ) = a , f ( c ) = b , f ( d ) = c$. Find the inverse of $f$.

Accepted Answer

The answer of Suppose that $f$ is the function from...

Question 15

Let $P ( m , n )$ be " $n$ is greater than or equal to $m$ " where the domain (universe of discourse) is the set of nonnegative integers. What are the truth values of $\exists n \forall m P ( m , n )$ and $\forall m \exists n P ( m , n )$ ?

Accepted Answer

For every positive integer @#LAT-DLM& th

Question 16

Let $A = \{ a , c , e , h , k \} , B = \{ a , b , d , e , h , i , k , l \}$, and $C = \{ a , c , e , i , m \}$. Find each of the following sets.
(a) $A \cap B$
(b) $A \cap B \cap C$
(c) $A \cup C$
(d) $A \cup B \cup C$
(e) $A - B$
(f) $A - ( B - C )$

Accepted Answer

(a) @#LAT-DLM&
(b) @#LAT-DLM&

Question 17

Show that ¬(p ∨ ¬q) and q ∧ ¬p are equivalent 
(a) using a truth table. 
(b) using logical equivalences.

Accepted Answer

(a) We have the following truth table. @#IMG-DLM&

Question 18

Prove or disprove that (p → q) → r and p → (q →r) are equivalent.

Accepted Answer

Suppose that p is false, q is true, and

Exam 1: The Foundations: Logic and Proofs

Prove that all the solutions to the equation x2=x+1 are irrational. \text { Prove that all the solutions to the equation } x ^ { 2 } = x + 1 \text { are irrational. } Prove that all the solutions to the equation x2=x+1 are irrational.

What is the truth value of (p ∨ q) → (p ∧ q) when both p and q are false?

Prove that there are no solutions in positive integers to the equation x4 + y4 = 100.

Write the converse and contrapositive of the statement "If it is sunny, then I will go swimming."

Prove each of the following statements. (a) The sum of two even integers is always even. (b) The sum of an even integer and an odd integer is always odd.

Suppose that Q(x) is the statement "x + 1 = 2x." What are the truth values of ∀x Q(x) and ∃x Q(x)?

(a) Prove or disprove that a 6×66 \times 66×6 checkerboard with four squares removed can be covered with straight triominoes. (b) Prove or disprove that an 8×88 \times 88×8 checkerboard with four squares removed can be covered with straight triominoes.

Let f(n)=2n+1f ( n ) = 2 n + 1f(n)=2n+1 . Is fff a one-to-one function from the set of integers to the set of integers? Is fff an onto function from the set of integers to the set of integers? Explain the reasons behind your answers.

Find the values of ∑j=11002 and ∑j=1100(−1)j. \text { Find the values of } \sum _ { j = 1 } ^ { 100 } 2 \text { and } \sum _ { j = 1 } ^ { 100 } ( - 1 ) ^ { j } \text {. } Find the values of ∑j=1100​2 and ∑j=1100​(−1)j.

Suppose that fff is the function from the set {a,b,c,d}\{ a , b , c , d \}{a,b,c,d} to itself with f(a)=d,f(b)=a,f(c)=b,f(d)=cf ( a ) = d , f ( b ) = a , f ( c ) = b , f ( d ) = cf(a)=d,f(b)=a,f(c)=b,f(d)=c . Find the inverse of fff .

Show that ¬(p ∨ ¬q) and q ∧ ¬p are equivalent (a) using a truth table. (b) using logical equivalences.

Prove or disprove that (p → q) → r and p → (q →r) are equivalent.

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

A: Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

$\text { Prove that all the solutions to the equation } x ^ { 2 } = x + 1 \text { are irrational. }$

Prove that there are no solutions in positive integers to the equation x⁴ + y⁴ = 100.

(a) Prove or disprove that a $6 \times 6$ checkerboard with four squares removed can be covered with straight triominoes. (b) Prove or disprove that an $8 \times 8$ checkerboard with four squares removed can be covered with straight triominoes.

Let $f ( n ) = 2 n + 1$ . Is $f$ a one-to-one function from the set of integers to the set of integers? Is $f$ an onto function from the set of integers to the set of integers? Explain the reasons behind your answers.

$\text { Find the values of } \sum _ { j = 1 } ^ { 100 } 2 \text { and } \sum _ { j = 1 } ^ { 100 } ( - 1 ) ^ { j } \text {. }$

Suppose that $f$ is the function from the set $\{ a , b , c , d \}$ to itself with $f ( a ) = d , f ( b ) = a , f ( c ) = b , f ( d ) = c$ . Find the inverse of $f$ .