A binder includes a cross-linked product of at least a first polymer, a second polymer, and a third polymer, wherein the cross-linked product is cross-linked by at least two ester bonds; the first polymer includes polyimide, polyamic acid, a copolymer thereof, or a combination thereof, wherein the first polymer includes a structural unit including an alkali metal and a structural unit including at least one hydroxyl functional group; the second polymer includes poly(acrylic acid), poly(methacrylic acid), a copolymer thereof, or a combination thereof; and the third polymer includes polyvinyl alcohol, polyacrylamide, polymethacrylamide, a copolymer thereof, or a combination thereof.