1. **Problem Statement:** Construct a regular expression (RE) for the language over the alphabet $\Sigma = \{a,b,c\}$ that accepts all strings containing at least one $a$ followed by at least one $b$ followed by at least one $c$. 2. **Understanding the problem:** We want strings where somewhere in the string there is a sequence of one or more $a$s, immediately followed by one or more $b$s, immediately followed by one or more $c$s. 3. **Key points:** - The string can have any characters from $\{a,b,c\}$ before or after this sequence. - The sequence $a^+b^+c^+$ must appear at least once. 4. **Regular Expression construction:** - $a^+$ means one or more $a$s: $a^+ = aa^* = a a^*$ - $b^+$ means one or more $b$s: $b^+ = bb^* = b b^*$ - $c^+$ means one or more $c$s: $c^+ = cc^* = c c^*$ 5. **Allow any characters before and after:** - Use $(a|b|c)^*$ to represent any string over $\Sigma$. 6. **Final RE:** $$ (a|b|c)^* a^+ b^+ c^+ (a|b|c)^* $$ This RE matches any string over $\{a,b,c\}$ that contains at least one $a$ followed by at least one $b$ followed by at least one $c$ somewhere inside it. **Answer:** $$ (a|b|c)^* a^+ b^+ c^+ (a|b|c)^* $$