Chapter 9-11 Discrete Math Final Study Guide

The shape ABC is located at (1,1), (4,0), and (-1,-1). Using matrices, rotate this shape 180 degrees in the counterclockwise direction. What are the coordinates for A'B'C? (1,1), (-4,0), (1,1) (-1,-1), (-4,0), (-1,-1) (1,1), (-4,0), (-1,-1) (-1,-1), (-4,0), (1,1)

(-1,-1), (-4,0), (1,1)

The shape ABC is located at (1,1), (4,0), and (-1,-1). Using matrices, rotate this shape 90 degrees in the counterclockwise direction. What are the coordinates for A'B'C'? (-1,1), (4,0), (1,-1) (1,1), (0,4), (1,-1) (-1,1), (0,4), (1,1) (-1,1), (0,4), (1,-1)

(1,1), (0,4), (1,-1)

If point A is found at (1,3) and is translated 6 units up, then the new coordinates would be: (1,9) (7,9) (1,-3) (7,3)

(1,9)

A statement that is True regardless of the truth values of all the variables in it is called a tautology. Choose the tautology from the list below. (NOT q) OR q (p AND p) OR p q AND (q OR NOT q) p AND NOT p

(NOT q) OR q

A statement that is True regardless of the truth values of all the variables in it is called a tautology. Choose the tautology from the list below. (p AND p) OR p (NOT q) OR q q AND (q OR NOT q) p AND NOT p

(NOT q) OR q = 1. (tautology)

Let m stand for 'Math is fun', n stand for 'Today is Thursday', and p stand for 'It is raining.' Write a Boolean expression for the statement 'Today is Thursday and it's not raining, or math is fun.' (n V¬ p) ^ m (n ^ p) V m (n ^ ¬p) ^ m (n ^ ¬p) V m

(n ^ ¬p) V m

Let m stand for 'Math is fun', n stand for 'Today is Thursday', and p stand for 'It is raining.' Write a Boolean expression for the statement 'Today is Thursday and it's not raining, or math is fun.' (n v ¬p) ^ m (n ∧ ¬p) ^ m (n ∧ p) ∨ m (n ∧ ¬p) ∨ m

(n ∧ ¬p) ∨ m

When ordering binary numbers in a Karnaugh Map, which of the following is the correct sequence? 00, 01, 10, 11 11, 10, 01, 00 00, 01, 11, 10 00, 11, 10, 01

00, 01, 11, 10

The identity matrix can be likened to what number? 10 2 0 20 1

1

Which truth table output corresponds to the following circuit? 1, 0, 0, 1 0, 1, 1, 0 None of these. 1, 1, 1, 0

1, 0, 0, 1

How is 3 represented as information in a computer? 0 1 0 1 0 1 1 2 1 1 1 1 1

11

Where does Boolean logic derive its name from? The Boole Tree, a giant sequoia tree species 19th century mathematician George Boole The International Committee on Boolean Logic The Latin term for the concept of 'two opposites'

19th century mathematician George Boole

Where does Boolean logic derive its name from? The International Committee on Boolean Logic 19th century mathematician George Boole The Boole Tree, a giant sequoia tree species The Latin term for the concept of 'two opposites'

19th century mathematician George Boole

Which of the following combinations of gates would produce the following truth table? x y ? 0 0 1 0 1 0 1 0 0 1 1 1

1st pic

How many different values can a Boolean represent? 0 2 any integer 1

2

How many different values can a Boolean represent? 1 0 any integer 2

2

Which number needs to be changed to find the inverse of this matrix? [ 1 0 0] [0 1 2] [0 0 1]

2

Which number needs to be changed to find the inverse of this matrix? [1 0 0] [0 1 2] [0 0 1]

2

How many rows would a matrix that represents a system of 5 variables and 3 equations have? 6 5 3 8 2

3

When you multiply matrices, if the first matrix is a 4x3 matrix, which of the following matrices can we multiply with the first? 2x3 5x3 4x3 3x5

3 x 5

What is the number in the first row and first column in the answer matrix? [ 4 2 4] [1 2 ] [5 8 0] [ 2 0 ] [ 3 1]

4 x 1 ,2 x 2, 4 x 3 add the values = 20

Which system of equations is represented by the following matrix? (Think of the vertical line as the equal sign.) 4 -1 1 10 5 8 7 -5 6 8 -2 1

4x - y + z = 10 5x + 8y + 7z = -5 6x + 8y - 2z = 1

What is the number in the second row and third column in the answer matrix? [4 2 4] [1 2 6] [5 8 0] + [2 0 7]

7

The rotation matrix for 270 degrees in the counterclockwise direction is the same one as the one for a: 360 degrees clockwise rotation 270 degrees clockwise rotation 180 degrees clockwise rotation 90 degrees clockwise rotation

90 degrees clockwise rotation

A matrix is: Ways by which a shape, line, or point can be manipulated A rectangular array of numbers, symbols, or functions A graphical representation of numbers, symbols, or functions A scenario where every point in a figure is moved the exact same distance and in the same exact direction

A rectangular array of numbers, symbols, or functions

When you use the advanced settings in an online search engine and use the option to search for 'all these words', which type of Boolean logic are you using? NOT AND OR No Boolean logic is used

AND

Logic circuits can be used for: outputting a 'one' when the input bit is a Fibonacci number creating a CPU designing a simple voting system All of these answers are correct.

All of these answers are correct.

Adding an additional edge to a spanning tree would produce _____. a loop a cycle a redundant path All of these are correct.

All of these are correct.

Spanning trees are used in computer networks to assure _____. All of these are correct. A minimum number of links are used. There are no cycles or loops. All computers are available on the network.

All of these are correct.

When you use the advanced settings in an online search engine and use the option to search for 'all these words', which type of Boolean logic are you using? OR No Boolean logic is used AND NOT

And

What does ALU stand for? Arithmetic logic unit Arithmetic language unit Array logic unit Arithmetic logic unicode

Arithmetic logic unit

Karnaugh Maps are useful for Boolean expressions of how many variables? 1 variable Between 2 and 6 1 or 2 variables More than 6 Worksheet Answers

Between 2 and 6

How are logic circuits built? By keeping the expression as complex as possible By combining logic gates according to the function expression By simplifying the circuit using Karnaugh maps By counting the 1s and 0s in the expression

By combining logic gates according to the function expression

The operation OR is also known as: negation disjunction conjuction injunction

Disjunction

Which of the following is NOT an aim of Boolean circuits simplification? Less expensive circuit Dissipate more heat Consume less power Add more functionality on the chip

Dissipate more heat

Which of the following is not a valid matrix operation? Multiplication Subtraction Division Addition

Division

Why are NAND and NOR gates considered to be functionally complete? They are equivalent to each other. They require fewer transistors. Each could be made of AND, OR and NOT gates. Each could be used exclusively to form any other gate operation.

Each could be used exclusively to form any other gate operation.

The Boolean NOT operator: Always results in a 0. Ignores the state (1 or 0) it applies to. Flips the state (1 or 0) it applies to. Always results in a 1.

Flips the state (1 or 0) it applies to.

Let's say we have a variable value called ''rainy_night''. Now let's say you want to stay in your cozy house, but you don't want to go when it's warm and clear, just when it's a raining. You can use Boolean logic: IF (rainy_night) WE... IF (rainy_night) THERE... IF (rainy_night) THEN... IF (rainy_night) SO...

IF (rainy_night) THEN...

What does XOR mean? If P and Q are both true, the result is true. Otherwise, the result is false. If either P or Q are true, the result is true. Otherwise, the result is false. To flip the current state. True becomes false and false becomes true. If either P or Q is true--but not both--the output is true. Otherwise, the result is false.

If either P or Q is true--but not both--the output is true. Otherwise, the result is false.

Which of these is not true about Boolean logic? It evaluates whether something is 1 or 0. It evaluates whether something is TRUE or FALSE. It makes a complicated set of conditions much easier to evaluate. It evaluates whether integers are TRUE or FALSE.

It evaluates whether integers are TRUE or FALSE.

When using a Karnaugh Map to simplify an expression, what do you look for? A: Groupings of the 1 output either in a straight line or forming a square or rectangle B: Random locations of the 1 output Both A and B Just B Neither A nor B Just A

Just A

A two-dimensional rectangular array of numbers representing a shape's coordinates is properly called a: Matrix Rotation Transformation Neo

Matrix

A matrix B multiplied by its identity matrix is equal to what? Matrix B times 3 Matrix B squared Matrix B Its inverse Matrix B times 2

Matrix B

Which formula is equivalent to an OR gate? NOR(X,NOR(NOR(X,Y),NOR(X,Y))) NOR(NOR(X,Y),Y) NOR(NOR(X,X),NOR(Y,Y)) NOR(NOR(X,Y),NOR(X,Y))

NOR(NOR(X,Y),NOR(X,Y))

Which gate is equivalent to the circuit shown? XOR OR NOT NAND

NOT

Simplify the Boolean expression as much as possible. (p v¬q) ^ (¬p V ¬q)

NOT q

Simplify the Boolean expression as much as possible. (p V¬ q) ^ (¬ p V ¬ q) NOT q NOT p p q Worksheet Answers

Not q

What is a logic gate? Electronic circuitry which control the flow of electrical current based on logic A transistor switch in the open position One or more transistor switches used to convert a binary input into a binary output based on a logical operation A logical analysis of the input and outputs for a calculation performed by a computer's central processing unit

One or more transistor switches used to convert a binary input into a binary output based on a logical operation

What's a simplified Boolean expression for this Karnaugh Map? Output = NOT c Output = a & b & c Output = c Output = (NOT a & NOT b & c)

Output = c

Finding the minimum spanning tree by starting at a random node and adding the node with the lowest weight link is called _____. Prim's Algorithm Kruskal's Algorithm explicit enumeration tree traversal

Prim's Algorithm

Which of the following is a practical method to simplify Boolean functions of more than 6 variables? Boolean algebra Truth table Quine-McCluskey Karnaugh maps

Quine-McCluskey

The turning of a figure or object about a fixed point is called a: Reflection Rotation Dilation Translation

Rotation

A Karnaugh map does what to Boolean expressions? Translates them Complicates them Simplifies them Processes them

Simplifies them

What kinds of brackets are used in proper matrix notation? Absolute value brackets Sigma brackets Curly brackets Square brackets

Square brackets

What are transistor switches? Switches that manipulate binary numbers by opening and closing gates. Switches that connect the ALU to the CPU. Switches the send electrical currents through the ALU. Switches that speed up the processing of data in a computer.

Switches that manipulate binary numbers by opening and closing gates.

A matrix multiplied by its inverse is equal to what? The identity matrix. The matrix times 2. The matrix squared. A random matrix. The matrix times 3.

The identity matrix

The simplified expression of the following Boolean function: Z = ab'c'd' + abc'd' + a'b'cd' + ab'cd' + a'bcd' + abcd' + ab'c'd + abc'd + a'b'cd + ab'cd + a'bcd + abcd is Z = a + c. Which of these statements is FALSE The only way to simplify this is with a Quine-McCluskey algorithm. There will be 12 AND gates in the circuit diagram. Simplifying the circuit has practical applications such as making the circuit less expensive to build and consuming less power when it is utilized. This function can be simplified by using a Karnaugh map.

The only way to simplify this is with a Quine-McCluskey algorithm.

Which row would you multiply by -2 to help you find the inverse matrix? [ 1 0 0] [0 1 2] [0 0 1]

Third

What does a computer use to store binary numbers? Transistor switches in the open or closed position Magnetic storage elements with opposite orientations Optical switches with flat areas or bumps Electrical circuits with positive and negative charges

Transistor switches in the open or closed position

A _____ is a scenario where every point in a figure is moved the exact same distance and in the same exact direction. Translation Transformation Reflection Rotation

Translation

The _____ represents the values by which the _____ must be moved. Coordinate matrix, translation matrix Translation matrix, coordinate matrix Translation matrix, rotation matrix Reflection matrix, translation matrix

Translation matrix, coordinate matrix

Consider the following inputs: c = 5; b = 8; d = 3. What does the following expression return? (c < b) OR (b < d) True 3 Invalid False

True

Consider the following inputs: c = 5; b = 8; d = 3. What does the following expression return? (c < b) OR (b < d) True False 3 Invalid

True

Which of the following Boolean expressions returns False? True AND True True OR False True AND False NOT False

True AND False

Which of the following Boolean expressions returns False? True OR False True AND False True AND True NOT False

True AND False

Transformations in math are: A rectangular array of numbers, symbols, or functions A graphical representation of matrix coordinates Ways by which a shape, line, or point can be manipulated A scenario where every point in a figure is moved the exact same distance and in the same exact direction

Ways by which a shape, line, or point can be manipulated

Use De Morgan's Law to rewrite the negation of the statement: You get home by 10pm or you're grounded. You get home by 10pm or you're not grounded. You don't get home by 10pm and you're not grounded. You don't get home by 10pm and you're grounded. You didn't get home by 10pm and you're not not grounded.

You don't get home by 10pm and you're grounded. You don't get home by 10pm and you're not grounded. You get home by 10pm or you're not grounded. You didn't get home by 10pm and you're not not grounded.

Use De Morgan's Law to rewrite the negation of the statement: You get home by 10pm or you're grounded. You get home by 10pm or you're not grounded. You don't get home by 10pm and you're grounded. You don't get home by 10pm and you're not grounded. You didn't get home by 10pm and you're not not grounded.

You don't get home by 10pm and you're not grounded.

Which of these symbols is commonly used to enclose a matrix? ! [ ] { } Σ

[ ]

Subtract the following matrices: [1 -2] [1 -3] [4 -6] [3 -9] [0 1] [1 3] [-2 -1] [-3 -3] [-1 0] [-1 -3] [-1 0] [-1 -3] [-3 4] [-2 -6] [-1 -2] [-2 -6]

[0 1] [1 3]

Add the following matrices: [2 1] + [-4 12] [-2 -4] [2 14]

[3 8] [-2 16]

Examples of a spanning tree would NOT include _____. a calling tree a shopping list an organizational chart a computer network

a shopping list

What is an example of a generalized tree in discrete math? a tree with loops a simple spanning tree all of these are correct a non-spanning tree

all of these are correct

A decision tree has only two options at each node. What is this called? Binary decision tree Two-noded decision tree Two-edged decision tree Two options at each node is not a possibility in decision trees.

binary decision tree

Which of the following circuits is the simplest that corresponds to the following function: Z= abc'+abc ?

c b a Ω

What is a database NOT required to do? Preserve Data Retrieve Data. Correct Data. Store Data.

correct data

The operation OR is also known as: injunction conjuction negation disjunction

disjunction

A spanning tree connects nodes to the network _____. multiple ways from a single central node exactly once exactly twice

exactly once

Finding the minimum spanning tree by listing all possible spanning trees, totaling their weights, and selecting the tree with the lowest total weight is called _____. Prim's Algorithm explicit enumeration tree traversal Kruskal's Algorithm

explicit enumeration

Deleting unused data is called _____. data compression data recycling linear sorting garbage collection

garbage collection

Searching from left to right is a(n) _____ search strategy. postorder reorder preorder inorder

inorder

A spanning tree for n nodes uses _____ edges. n n + 1 n(n - 1) n - 1

n -1

The number of comparisons required to sort n items into a tree structure varies as _____. n-squared nlog(n) n n-cubed

n-log(n)

The number of comparisons required to sort n items linearly varies as _____. n-squared n n-cubed nlog(n)

n-squared

In a tree structure, the point at which a discrete decision is made is called a _____. node block root corner

node

Work to sort, organize the data, or maintain the Tree Sort is referred to as _____. system margin overhead sunk costs through-put

overhead

A search which starts at the tips (leaves) of the tree and works toward the root is the _____ strategy. reorder postorder inorder preorder

postorder

A search which starts at the root of the tree and bears left is the _____ strategy. postorder reorder preorder inorder

preorder

Which of the following is not a tree search strategy? Preorder Postorder Inorder Reorder

reorder

The decision tree structure begins at a node called the _____. branch top root span

root

If it is not possible to simplify a function using Karnaugh maps, this means that the function cannot be simplified in any other methods the function might be simplified by another method or it is in its simplest form The function is erroneous the function is too difficult to be simplified

the function might be simplified by another method or it is in its simplest form

A minimum spanning tree is the tree with _____. the minimum total weights no total weights the most symmetric total weights the maximum total weights

the minimum total weights

Design a logic circuit means transfering a human-readable specification into a circuit diagram deducting a human readable specification from a circuit or a circuit diagram simplifying a logic function using Karnaugh maps analyzing the circuit

transfering a human-readable specification into a circuit diagram

Data sorted in a tree structure may occasionally need to be resorted to maintain _____. tree symmetry entropy dynamic tension tree asymmetry

tree symmetry

What is NOT a step of the systematic procedure of designing a logic circuit? simplify the truth table find the function and draw the circuit find the truth table use different wire colors to connect the components

use different wire colors to connect the components

Discrete math deals with problems concerning options that can be expressed as _____. odd numbers whole numbers fractions all real numbers

whole numbers

A spanning tree must connect nodes _____. with a minimum number of edges from top down from bottom up symmetrically

with a minimum number of edges