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Discrete Maths Questions, Exercises of Discrete Mathematics

Institute of Business Administration (IBA)Discrete Mathematics

Covers all the basics of the first 2 chapters of Susanna Book for Discrete Maths

Typology: Exercises

2023/2024

Uploaded on 05/10/2024

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arbaaz-murtaza-2102208 🇵🇰

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Discrete Mathematics
Assignment#02
1. Use truth tables to determine whether the argument form is valid:
p∨q
p−→∼ q
p−→ r
∴r
2. Use symbols to write the logical form of each argument. If the argument is valid, identify
the rule of inference that guarantees its validity. Otherwise, state whether the converse
or the inverse error is made.
a)
If Jules solved this problem correctly, then Jules obtained the answer 2.
Jules obtained the answer 2.
∴Jules solved this problem correctly.
b)
This real number is rational or it is irrational.
This real number is not rational.
∴This real number is irrational.
3. A set of premises and a conclusion are given. Use the valid argument forms to deduce
the conclusion from the premises, giving a reason for each step. Assume all variables are
statement variables.
4. Find the truth set of each predicate.
a) predicate: 6/d is an integer, domain: Z
c) predicate: 1 ≤x2≤4, domain: R
5. Let the domain of x be the set D of objects discussed in mathematics courses, and let
Real(x) be “x is a real number,” Pos(x) be “x is a positive real number,” Neg(x) be “x
is a negative real number,” and Int(x) be “x is an integer.”
a. ∀x, Real(x)∧N eg(x)−→ P os(−x).
b. ∃xsuch that Real(x)∧ ∼ Int(x).
pf3

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Discrete Mathematics

Assignment#

  1. Use truth tables to determine whether the argument form is valid: p ∨ q p −→∼ q p −→ r ∴ r
  2. Use symbols to write the logical form of each argument. If the argument is valid, identify the rule of inference that guarantees its validity. Otherwise, state whether the converse or the inverse error is made. a) If Jules solved this problem correctly, then Jules obtained the answer 2. Jules obtained the answer 2. ∴ Jules solved this problem correctly. b) This real number is rational or it is irrational. This real number is not rational. ∴ This real number is irrational.
  3. A set of premises and a conclusion are given. Use the valid argument forms to deduce the conclusion from the premises, giving a reason for each step. Assume all variables are statement variables.
  4. Find the truth set of each predicate. a) predicate: 6/d is an integer, domain: Z c) predicate: 1 ≤ x^2 ≤ 4, domain: R
  5. Let the domain of x be the set D of objects discussed in mathematics courses, and let Real(x) be “x is a real number,” Pos(x) be “x is a positive real number,” Neg(x) be “x is a negative real number,” and Int(x) be “x is an integer.” a. ∀x, Real(x) ∧ N eg(x) −→ P os(−x). b. ∃x such that Real(x)∧ ∼ Int(x).
  1. write a negation for each statement: a) ∀ real numbers x, if x^2 ≥ 1 then x > 0. b)∀ integers a, b and c, if a − b is even and b − c is even, then a − c is even. c) If the square of an integer is odd, then the integer is odd.
  2. For the statement write the converse, inverse, and contrapositive. Indicate as best as you can which among the statement, its converse, its inverse, and its contrapositive are true and which are false. Give a counterexample for each that is false. ∀ real numbers x, if x^2 ≥ 1 then x > 0.
  3. Use the facts that the negation of a ∀ statement is a ∃ statement and that the negation of an if − then statement is an and statement to rewrite each of the statements without using the word necessary or suf f icient. a) Being divisible by 8 is not a necessary condition for being divisible by 4. b) Having a large income is not a sufficient condition for a person to be happy.
  4. Indicate which of the following statements are true and which are false. Justify your answers as best you can. a) ∀x ∈ Z, ∃y ∈ Z such that x = y + 1. b) ∀x ∈ R, ∃y ∈ R such that xy = 1.
  5. (a) Rewrite the statement formally using quantifiers and variables, and (b) write a negation for the statement.
  1. Somebody loves everybody.
  2. Any even integer equals twice some integer.
  1. The notation ∃! stands for the words “there exists a unique.” Thus, for instance, ”∃!x such that x is prime and x is even” means that there is one and only one even prime number. Which of the following statements are true and which are false? Explain. a) ∃! real number x such that ∀ real numbers y, xy = y. b) ∃! integer x such that 1/x is an integer. c) ∀ real numbers x, ∃! real number y such that x + y = 0. 12..
  2. Let D = E = {− 2 , − 1 , 0 , 1 , 2 }. Write negations for each of the following statements and determine which is true, the given statement or its negation. a) ∀x in D, ∃y in E such that x + y = 1. b) ∃x in D such that ∀y in E, x + y = −y.

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