Conditionals and Translations
CSI 2101: Discrete Structures
School of Electrical Engineering and Computer Science, University of Ottawa
January 21, 2022
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1 Equivalence of Conditional Statements
2 Translations and Quantification
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Equivalence of Conditional Statements
Show that (p → q) ∧ (p → r) and p → (q ∧ r) are logically equivalent.
Show that (p → q) ∨ (p → r) and p → (q ∨ r) are logically equivalent.
Show that ¬p → (q → r) and q → (p ∨ r) are logically equivalent.
Showthat(p∧q)→r and(p→r)∧(q→r)arenotlogically equivalent.
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Mathematical to Nested Quantifiers
(** Textbook examples from 1.5)
Translate the statement “The sum of two positive integers is always
positive” into a logical expression.
Translate the statement “Every real number except zero has a multiplicative inverse” into a logical expression.
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Nested Quantifiers to English
(** Textbook examples from 1.5)
Translate the statement ∀x(C(x)∨∃y(C(y)∧F(x,y))) into English, where C(x) is “x has a computer,” F(x,y) is “x and y are friends,” and the domain for both x and y consists of all students in your school.
Translate the statement
∃x ∀y ∀z ((F (x , y ) ∧ F (x , z ) ∧ (y ̸= z )) → ¬F (y , z )) into English, where F(a,b) means a and b are friends and the domain for both x, y and z consists of all students in your school.
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From English to Logical
(** Textbook examples from 1.5)
Express the statement “If a person is female and is a parent, then this person is someone’s mother” as a logical expression involving predicates, quantifiers with a domain consisting of all people, and logical connectives.
Express the statement “Everyone has exactly one best friend” as a logical expression involving predicates, quantifiers with a domain consisting of all people, and logical connectives.
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Thank You!
Questions and Comments?
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