UGRD-CS6105 Discrete Mathematics
Prelim Q1 to Prelim Exam, Midterm Q1, Q2, Finals Q1, Q2
Classify the sentence below as an atomic statement, a molecular statement, or not a statement at all. If the statement is molecular, identify what kind it is (conjuction, disjunction, conditional, biconditional, negation)
Everybody needs somebody sometime.
-Atomic N/A
Classify the sentence below as an atomic statement, a molecular statement, or not a statement at all. If the statement is molecular, identify what kind it is (conjuction, disjunction, conditional, biconditional, negation).
We can have donuts for dinner, but only if it rains.
-Molecular Conditional
Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}
Find A ∩ B
-{3, 4, 5}
Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}
Find A \ B
-{1, 2}
Determine whether the sentence below is an atomic statement, a molecular statement, or not a statement at all.
• The customers wore shoes.
-Molecular
Classify the sentence below as an atomic statement, a molecular statement, or not a statement at all. If the statement is molecular, identify what kind it is (conjuction, disjunction, conditional, biconditional, negation).
Every natural number greater than 1 is either prime or composite.
-Molecular Conditional
Find the cardinality of R = {20,21,...,39, 40}
| R | = 21
Find the cardinality of S = {1, {2,3,4},0}
| S | = 3
Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}
Find A U B
-{1, 2, 3, 4, 5, 6, 7}
Classify the sentence below as an atomic statement, a molecular statement, or not a statement at all. If the statement is molecular, identify what kind it is (conjuction, disjunction, conditional, biconditional, negation). The sum of the first 100 odd positive integers.
-Atomic N/A
In my safe is a sheet of paper with two shapes drawn on it in colored crayon. One is a square, and the other is a triangle. Each shape is drawn in a single color. Suppose you believe me when I tell you that if the square is blue, then the triangle is green. What do you therefore know about the truth value of the following statement?
The square and the triangle are both green. - The statement is FALSE
Find | R | when R = {2, 4, 6,..., 180}
- 90
Let A = {3, 4, 5}. Find the cardinality of P(A)
- 8
In my safe is a sheet of paper with two shapes drawn on it in colored crayon. One is a square, and the other is a triangle. Each shape is drawn in a single color. Suppose you believe me when I tell you that if the square is blue, then the triangle is green. What do you therefore know about the truth value of the following statement?
The square and the triangle are both blue. - The statement is FALSE
The cardinality of {3, 5, 7, 9, 5} is 5.
-False
n my safe is a sheet of paper with two shapes drawn on it in colored crayon. One is a square, and the other is a triangle. Each shape is drawn in a single color. Suppose you believe me when I tell you that if the square is blue, then the triangle is green. What do you therefore know about the truth value of the following statement?
If the triangle is not green, then the square is not blue. - The statement is TRUE
Determine whether the sentence below is an atomic statement, a molecular statement, or not a statement at all.
• Customers must wear shoes. - Not a Statement
Find |A ∩ B| when A = {1, 3, 5, 7, 9} and B {2, 4, 6, 8, 10}
- 0
Classify the sentence below as an atomic statement, a molecular statement, or not a statement at all. If the statement is molecular, identify what kind it is (conjuction, disjunction, conditional, biconditional, negation).
The Broncos will win the Super Bowl or I’ll eat my hat.
-Molecular Conjunction
GIven the function :
f : Z → Z defined by f = 3n
Which of the following is a possible range of the function?
-all multiples of three
For a function f : N → N, a recursive definition consists of an initial condition together with a recurrence relation.
Find f (1).
- 4
In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?
- 720
How many 3-letter words with or without meaning, can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed?
- 720
Consider the function f : N → N given by f (0) 0 and f (n + 1) f + 2n + 1. Find f (6).
- 36
Rule that states that every function can be described in four ways: algebraically (a formula), numerically (a table), graphically, or in words.
- Rule of four
Defined as the product of all the whole numbers from 1 to n.
-Factorial
Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?
- 210
IN combinations, the arrangement of the elements is in a specific order.
-false
The ________________________ states that if event A can occur in m ways, and event B can occur in n disjoint ways, then the event “A or B” can occur in m + n ways.
-Additive principle
Answer the following: f (1) = 4
What is the element n in the domain such as f = 1 2
Find an element n of the domain such that f = n. 3
Additive principle states that if given two sets A and B, we have |A × B| |A| · |B
-false
Given the diagram, answer the following questions :
How many people takes tea and wine? 32
How many people takes coffee but not tea and wine? 45
What is the difference of persons who take wine and coffee to the persons who the persons who takes tea only? 15
Let A, B and C represent people who like apples, bananas, and carrots respectively. The number of people in A = 10, B = 12 and C = 16. Three people are such that they enjoy apples, bananas as well as carrots. Two of them like apples and bananas. Let three people like apples and carrots. Also, four people are such that they like bananas and carrots.
How many people like apples only? 2
How many people like only one of the three? 26
It is a rule that assigns each input exactly one output
-Function
Determine the number of elements in A U B.
- 18
In my safe is a sheet of paper with two shapes drawn on it in colored crayon. One is a square, and the other is a triangle. Each shape is drawn in a single color. Suppose you believe me when I tell you that if the square is blue, then the triangle is green. What do you therefore know about the truth value of the following statement?
If the triangle is green, then the square is blue. - The statement is TRUE
A bijection is a function which is both an injection and surjection. In other words, if every element of the codomain is the image of exactly one element from the domain
Consider the statement, “If you will give me a cow, then I will give you magic beans.” Determine whether the statement below is the converse, the contrapositive, or neither.
• You will give me a cow and I will not give you magic beans. - Converse
surjective and injecive are opposites of each other.
-false
The inverse image of a a subset B of the codomain is the set f −1 (B) {x ∈ X : f (x) ∈ B}.
The range is a subset of the codomain. It is the set of all elements which are assigned to at least one element of the domain by the function. That is, the range is the set of all outputs.
Which of the following translates into “Jack and Jill both passed math” into symbols?
-P Λ Q
In my safe is a sheet of paper with two shapes drawn on it in colored crayon. One is a square, and the other is a triangle. Each shape is drawn in a single color. Suppose you believe me when I tell you that if the square is blue, then the triangle is green. What do you therefore know about the truth value of the following statement?
The square is not blue or the triangle is green. - The statement is FALSE
Suppose P and Q are the statements:
P: Jack passed math.
Q: Jill passed math.
Translate "¬(P ν Q) → Q" into English.
-If Jack or Jill did not pass math, then Jill passed math.
Consider the statement, “If you will give me a cow, then I will give you magic beans.” Determine whether the statement below is the converse, the contrapositive, or neither.
• If I will give you magic beans, then you will give me a cow. - Neither
Consider the statement, “If you will give me a cow, then I will give you magic beans.” Determine whether the statement below is the converse, the contrapositive, or neither.
If I will not give you magic beans, then you will not give me a cow. - Contrapositive
Arithmetic progression is the sum of the terms of the arithmetic series.
-False
Given the series : 2,5,8,11....
What is the type of progression? Arithmetic
What is the sum from 1st to 5th element? 40
What is the missing term?
3,9,__,81....
- 27
Identify the propositional logic of the truth table given
-negation
The study of what makes an argument good or bad.
-Logic
match the following formulas to its corresponding sequence
Paths start and stop at the same vertex.