Bijections and Cardinality CS 2800: Discrete Structures, Spring 2015 Sid Chaudhuri Recap: Le t and Right !n"erses ● # function is injective $one-to-one) i' it has a left inverse – g : B → A is a left in"erse of f : A → B if g ( f (a) ) = a for all a ∈ A ● # function is surjective (onto) i' it has a right inverse – h : B → A is a right in"erse of f : A → B if f ( h (b) ) = b for all b ∈ B (hought for the Day #1 !s a left in"erse injecti"e or surjecti"e* +hy* !s a right in"erse injecti"e or surjecti"e* +hy* (Hint: how is f related to its left/right inverse?) Sur/injectivity o le t/right in"erses ● The left inverse is always surjective! – … since f is its right in"erse ● (he right in"erse is al-ays injecti"e. – / since f is its le t in"erse 0actoid for the Day #1 !f a function has both a left in"erse and a right in"erse, then the t-o in"erses are identical, and this com2on in"erse is unique (Prove!) (his is called the two-sided inverse, or usually just the inverse f –1 of the function f http:/,---.cs.cornell.edu,courses,cs2800,2015sp,handouts,jonpa56function_notes.pdf Bijection and two-sided in"erse ● # function f is bijecti"e i' it has a t-o-sided in"erse ● Proof (⇒): !f it is bijecti"e, it has a left in"erse $since injecti"e) and a right in"erse (since surjecti"e), which must be one and the sa2e by the previous factoid ● Proof (⇐): !f it has a t-o-sided in"erse, it is both injecti"e (since there is a left in"erse) and surjecti"e (since there is a right in"erse). 7ence it is bijecti"e. !n"erse o a unction ● (he in"erse of a bijecti"e function f : A → B is the unique function f ‑1: B → A such that for any a ∈ A, f ‑1(f(a)) = a and for any b ∈ B, f(f ‑1(b)) = b ● # function is bijecti"e i' it has an in"erse function f(a) a f ‑1(a) b = f(a) f A B f ‑1 0ollo-ing 8rnie Croot's slides !n"erse o a unction ● !f f is not a bijection, it cannot ha"e an in"erse function f x x 1 1 y ? y 2 2 z z 3 3 w w :nto, not one-to-one f -1(2) = ? 0ollo-ing 8rnie Croot's slides !n"erse o a unction ● !f f is not a bijection, it cannot ha"e an in"erse function f 1 x 1 x 2 2 y y 3 3 z ? z 4 4 :ne-to-one, not onto f -1(4) = ? 0ollo-ing 8rnie Croot's slides 7ow can we count elements in a set* 7ow can we count elements in a set* ● 8asy for finite sets – just count the elements. ● Does it e"en ma5e sense to ask a1out the num1er of elements in an infinite set* ● !s it meaningful to say one infinite set is larger than another* – #re the natural numbers larger than ● the e"en numbers* ● the rational numbers* ● the real numbers* 0ollo-ing 8rnie Croot's slides Cardinality and Bijections ● !f A and B are finite sets, clearly they ha"e the sa2e num1er of elements i' there is a bijection 1et-een them e.g. | {x, y, z} | = | {1, 2, 3} | = 3 x 1 y 2 z 3 0ollo-ing 8rnie Croot's slides Cardinality and Bijections ● Definition: Set A has the sa2e cardinality as set B, denoted |A| = |B|, i' there is a bijection from A to B – 0or finite sets, cardinality is the number of elements – (here is a bijection from n%element set A to {1, 2, 3, …, n } 0ollo-ing 8rnie Croot's slides Cardinality and Bijections ● =atural num1ers and e"en num1ers ha"e the sa2e cardinality 0 1 2 @ > ... 2 > ? 8 10 ... Sets having the same cardinality as the natural numbers (or some subset of the natural numbers) are called countable sets 0ollo-ing 8rnie Croot's slides Cardinality and Bijections ● =atural num1ers and rational num1ers ha"e the sa2e cardinality. 0 1 4 5 1 1 1 1 ⋯ 1 2 3 4 6 Illustrating proof 2 2 2 2 only for positive 1 2 3 rationals here, can 3 7 be easily extended 3 3 3 to all rationals 1 2 3 8 4 ⋱ 1 ⋮ Cardinality and Bijections ● (he natural num1ers and real num1ers do not ha"e the sa2e cardinality x1 0 . 0 0 0 0 0 0 0 0 0 … x2 0 . 1 0 3 0 4 0 5 0 1 … x3 0 . 9 8 7 6 5 4 3 2 1 … x4 0 . 0 1 2 1 2 1 2 1 2 … x5 ⁞ Cardinality and Bijections ● (he natural num1ers and real num1ers do not ha"e the sa2e cardinality Consider the number x 0 . 0 0 0 0 0 0 0 0 0 … 1 y = 0 . b1 b2 b3... x2 0 . 1 0 3 0 4 0 5 0 1 … 1 if the ith decimal place of x is zero b = i x3 0 . 9 8 7 6 5 4 3 2 1 … i 0 if it is non-zero x4 0 . 0 1 2 1 2 1 2 1 2 … y x x5 ⁞ cannot be equal to any i – it difers by one digit from each one. (here are many infinities (hought or the Day )2 Do the real inter"al [0, 1] and the unit square [0, 1] B [0, 1] ha"e the sa2e cardinality* 0,1 1,1 0 1 0,0 1,0 Comparing Cardinalities ● Definition: !f there is an injecti"e function from set A to set B, we say |A| ≤ |B| 2 > ? 8 ... 0 1 2 @ > 5 ? C 8 D ... |Evens| ≤ |N| 0ollo-ing 8rnie Croot's slides Comparing Cardinalities ● Definition: !f there is an injecti"e function from set A to set B, but not from B to A, we say |A| < |B| ● Cantor<SchrEder<Bernstein theorem: !f |A| ≤ |B| and |B| ≤ |A|, then |A| = |B| – Exercise: pro"e this. – i.e. sho- that there is a bijection from A to B if there are injective functions from A to B and from B to A – $it's not easy.& 0ollo-ing 8rnie Croot's slides.