Functions count the of functions f:[m→[m X one-one correspondence [n] [m] [m]→[m]台[m]n l[m→[m=lmlm|=mm "Combinatorial proof
Functions [n] [m] f : [n] [m] count the # of functions one-one correspondence [n] [m] ⇥ [m] n |[n] [m]| = |[m] n| = mn “Combinatorial proof
Injections count the of 1-1 functions f (n(m] one-one correspondence [n] [m] π=(f(1),f(2),.,f(n) n-permutation::元∈[m]of distinct elements m! (m)m=m(m-1).(m-n+1)= (m-n)！ “m lower factorial n
Injections [n] [m] count the # of 1-1 functions one-one correspondence [m] n of distinct elements = (f(1), f(2),...,f(n)) n-permutation: = m! (m n)! (m)n = m(m 1)···(m n + 1) “m lower factorial n” f : [n] 1-1 [m]
Subsets S={1,2,3} subsets of S: ☑， {I,2,{3}, {1,2,{1,3},{2,3, {1,2,3} S={1,c2,,xn} Power set:25-{TT S) 121=
Subsets S = { 1, 2, 3 } subsets of S: ∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} S = {x1, x2,...,xn} Power set: 2S = 2S = {T | T S}
Subsets S={c1,c2,,xn} Power set:25 ={T TCS) 121= Combinatorial proof: A subset TS corresponds to a string of n bit, where bit i indicates whether xiET
Subsets S = {x1, x2,...,xn} Power set: 2S = Combinatorial proof: A subset T S corresponds to a string of n bit, where bit i indicates whether xi ⇥ T. 2S = {T | T S}
Subsets S={x1,c2,.,xn} Power set:25 ={TT C S} |2=I{0,1}m1=2m Combinatorial proof: -日 c∈T φ：2s→{0,1}n ,庄T is 1-1 correspondence
Subsets S = {x1, x2,...,xn} Power set: 2S = Combinatorial proof: : 2S {0, 1}n (T)i = 1 xi T 0 xi ⇥ T is 1-1 correspondence |{0, 1}n| = 2n 2S = {T | T S}