Subsets subsets of 1,2,3 } 0, {1,{2},{3}, {1,2,{1,3},{2,3}, {1,2,3} [ml={1,2,.,n} Power set:2=SS C In]} |2=
Subsets subsets of { 1, 2, 3 }: ∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} Power set: [n] = {1, 2,...,n} 2[n] = 2[n] = {S | S ✓ [n]}
Subsets [m={1,2,.,n} Power set:2 m=SSCIn] |2网川= Combinatorial proof: A subset S n corresponds to a string of n bit, where bit i indicates whether ie s
Subsets Combinatorial proof: Power set: [n] = {1, 2,...,n} 2[n] = A subset S ✓ [n] corresponds to a string of n bit, where bit i indicates whether i 2 S. 2[n] = {S | S ✓ [n]}
Subsets [m={1,2,.,n} Power set: 2m=SSCIn]} 2则=K0,11=2m Combinatorial proof: S中gcm产an石：g one-to-one correspondence
Subsets |{0, 1}n| = 2n one-to-one correspondence [n] = {1, 2,...,n} 2[n] = S ✓ [n] S 2 {0, 1}n S(i) = ( 1 i 2 S 0 i 62 S 2[n] = {S | S ✓ [n]} Combinatorial proof: Power set:
Subsets [m={1,2,.,m} Power set: 2lm={S|S≤[n} 2= A not-so-combinatorial proof: Let f(n)=2mnl f(n)=2f(n-1)
Subsets A not-so-combinatorial proof: Let f(n)=2f(n 1) [n] = {1, 2,...,n} 2[n] = f(n) = 2[n] 2[n] Power set: = {S | S ✓ [n]}
r-2 f(n)=2f(n-1) 2m =isC In]Ings}U{S C In]lneS} 2=l2-刂+2n-20a- Sum rule: finite disjoint sets S and T SUT=S+T
f(n)=2f(n 1) Sum rule: finite disjoint sets S and T |S T| = |S| + |T| = 2f(n 1) f(n) = 2[n] 2[n] = 2[n] = 2[n1] 2[n1] + {S ✓ [n] | n 62 S} {S ✓ [n] | n 2 S}