I understand that $(0 U 1)^*$ is the set of all binary strings. However, what do the other two expressions create?Could someone please provide examples of what  strings these languages would accept?$L = \{0,1\}^*$$L = \{0,1\}^+$
$\{0,1\}^*$ vs $\{0,1\}^+$ vs $(0 \cup 1)^*$
formal languages;regular languages;regular expressions
There is some abuse of notation going on. There are regular expressions and there are sets of strings.$\{0, 1\}^*$ is the set of all binary strings. This is (rarely) used to mean the regular expression matching all binary strings.$(0 \cup 1)^*$ is the regular expression matching all binary strings.$(0 \cup 1)^+$ is the regular expression matching all non-empty binary strings.$\{0, 1\}^+$ is (rarely used) to denote the regular expression that matches all non-empty binary strings (same as $(0 \cup 1)^+$). It might be used in some places to denote the set of all non-empty binary strings, but I never personally saw this usage.