Need Help with Week 1 Practice Assignment Q. 12

[Kanhaiya Varshney]

Week 1 Practice Assignment Question 12 & 13
Really confused with this question even after checking the practice assignment video solution of this. 
Please help with a simple solution of how to form R1, R2, & R3 sets in this case.

Let S be the set {January, February, March, April, May, June, July, August, September, October, November, December} of months in a year. Define the following three relations:

• R1:={(a,b)|a,b∈S,R1:={(a,b)|a,b∈S, a and b have same last four letters. }
• R2:={(a,b)|a,b∈S,R2:={(a,b)|a,b∈S, a and b have same number of days. }
• R3:={(a,c)|a,c∈S,R3:={(a,c)|a,c∈S, for some b∈S,(a,b)∈R1,(b,c)∈R2}b∈S,(a,b)∈R1,(b,c)∈R2}

For example,(December, June)∈R3∈R3 since (December, September)∈R1∈R1 , (September, June)∈R2  

[Debajyoti Biswas]

Step 1: Write the names and number of days of all 12 months in a table

Step 2:  Write R1 : (Jan'uary', Jan'uary'), (Jan'uary', Febr'uary') etc. Only May will not fall in this set (as it has only 3 letters), otherwise all the months, will at least once be there. R1 = {(Jan, Jan), (Jan, Feb), (Feb, Jan), (Feb, Feb), (Mar, Mar), (April, April),(June, June), (July, July), (Aug, Aug), (Oct, Oct), (Sept, Sept), (Sept, Nov), (Sept, Dec),(Nov, Sept), (Nov, Nov), (Nov, Dec), (Dec, Sept), (Dec, Nov), (Dec, Dec)}

Step 3: Then you make another table, for R2, here some months have 31 days, some months have 30 days, February has 28 or 29 days.

So this set can be classified into 3 classes; class 1: those who have 31 days, class 2: those who have 30 days, class 3: February (28 or 29 days)

Step 4: Then you define R3. From the 3 classes that we assumed, 7 pairs will be there in R3 starting with January. These are {(Jan, Jan), (Jan, Mar), (Jan, May), (Jan, July), (Jan, Aug), (Jan, Oct), (Jan, Dec)}. Moreover, (Jan, Feb) is in R1, and Feb is in another partition in S due to R2. So there are total 8 pairs (adding (Jan,Feb) with previous 7 elements) in R3 starting with Jan. Likewise for all 12 months you need to calculate.

Step 5: Then you need to add the numbers of pairs.

Also please check the practice solution PDF that we uploaded.