230hw2sol - Math 230 E Fall 2011 Homework 2 Solutions Drew...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 230 E Fall 2011 Homework 2 Solutions Drew Armstrong Problem 1. Let x ⊕ y denote the XOR (exclusive or) function, which is defined by: 1 ⊕ 1 = 0 , 1 ⊕ 0 = 1 , ⊕ 1 = 1 , ⊕ 0 = 0 . Express x ⊕ y in terms of the basic operations ∧ , ∨ , ¬ . Express the usual “inclusive or” x ∨ y in terms of the operations ∧ , ⊕ , ¬ . Some trial-and-error is involved in this, but using Venn diagrams and the idea that x ⊕ y is the “same” as x Δ y can help a lot. Note that there are infinitely many correct answers, so yours may differ from mine. First, I claim that x ⊕ y = ¬ ( x ⇔ y ) = ¬ (( x ⇒ y ) ∧ ( y ⇒ x )) = ¬ ( ¬ ( x ∧¬ y ) ∧¬ ( y ∧¬ x )) = ( x ∧¬ y ) ∨ ( y ∧¬ x ) = various other things. To prove this I will use a truth table: x y x ⇔ y ¬ ( x ⇔ y ) x ⊕ y 1 1 1 1 0 1 1 0 1 1 1 0 0 1 Note that the last two columns are equal. The very last column was given to us by definition . Second, I claim that x ∨ y = ( x ⊕ y ) ⊕ ( x ∧ y ) = various other things. To prove this we again use a truth table: x y x ⊕ y x ∧ y ( x ⊕ y ) ⊕ ( x ∧ y ) x ∨ y 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 0 Note that the last two columns are equal. [FYI: The structure ( { , 1 } , ∧ , ∨ , ¬ ) is called a “Boolean algebra” — as we know — and the structure ( { , 1 } , ∧ , ⊕ , ¬ ) is called a “Boolean ring”. In a Boolean ring we can really think of ∧ as “multiplication mod 2 ” and ⊕ as “addition mod 2 ”.] Problem 2. Given two subsets S,T ⊆ U of some universal set, the statement “ S = T ” means exactly that “ ∀ x ∈ U,x ∈ S ⇔ x...
View Full Document

This note was uploaded on 01/08/2012 for the course MATH 461 taught by Professor Armstrong during the Fall '11 term at University of Miami.

Page1 / 3

230hw2sol - Math 230 E Fall 2011 Homework 2 Solutions Drew...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online