as shown below we need to give combinatorla proof and prove the four statements below. I have no clue.

U]. WlllCll lUdUb' LU [11111 ULPLUD'lUll U11 [1115 ngllh uuuu ElUU. 3. Give a combinatorial proof that '1 s '2 . 71+ 1 . , 3 3 1:0 3:0 . Hint: choose elements from the set {0, 1,2,. . . ,n}. 4. (For this question, you may use any clearly stated properties of strings and their concatenations, assuming they are true) We say that two regular expressions are equivalent if they have the same langauge. It turns out that regular expressions satisfy many familiar properties of addition and multiplication: (a) Show that there exists a regular expression m such that for all 'r, m + 7' w 7' bug x: this used to say m + r w m, it is correct now (b) Check that for all r, 7'0?" z m. (c) Give a regular expression 7'1 such that for all r E RE,rr1 w r. (d) Check that for all r, 3,15 6 RE, r(s + t) m rs + rt. (e) Show that there exists 1", s such that (r + s)* a 'r* + 5*