Question Video: Writing a Relation between Two Sets given Its Rule Mathematics • Third Year of Preparatory School

Video Transcript

For two sets 𝑋 and 𝑌, set 𝑋 contains the numbers one, five, six, and seven and set 𝑌 contains the numbers one, two, five, eight, nine, and 14. Determine the relation 𝑅 from 𝑋 to 𝑌, where 𝑎𝑅𝑏 means 𝑎 minus 𝑏 is a prime number, given 𝑎 is an element of 𝑋 and 𝑏 is an element of 𝑌.

We recall that a relation is a set of ordered pairs 𝑎, 𝑏. In this question, 𝑎 is an element of set 𝑋 and 𝑏 is an element of set 𝑌. We are told in this question that our ordered pairs must follow the rule that 𝑎 minus 𝑏 is a prime number. A prime number has exactly two factors. The first five of these are two, three, five, seven, and 11. As these are all positive and we are told that 𝑎 minus 𝑏 is a prime number, then 𝑎 must be greater than 𝑏. The number from set 𝑋 must be greater than the number from set 𝑌. This means that our value of 𝑎 cannot be one as there are no numbers in set 𝑌 that are less than one.

Let’s consider what can happen when 𝑎 is equal to five. Both one and two are less than five, which suggests our ordered pair could be five, one or five, two. Five minus one is equal to four, and five minus two is equal to three. Three is one of our prime numbers, whereas four is not. This means that the ordered pair five, one is not in the relation 𝑅. Let’s now consider what can happen when 𝑎 is equal to six. The numbers in set 𝑌, one, two, and five, are all less than six, which suggests we could have the ordered pairs six, one; six, two; or six, five.

Six minus one is equal to five, which is a prime number. Six minus two is equal to four. This is not a prime number as it is divisible by two. The ordered pair six, two is therefore not in the relation 𝑅. As six minus five is equal to one, which is not a prime number, the ordered pair six, five is not in the relation 𝑅.

The last element of set 𝑋 is seven, and the numbers one, two, and five in set 𝑌 are less than this. This means that our potential ordered pairs are seven, one; seven, two; and seven, five. Seven minus one is equal to six, which is not a prime number. Seven minus two is equal to five, and seven minus five is equal to two. Both of these answers are prime numbers.

We can, therefore, conclude that the relation 𝑅 has four ordered pairs: five, two; six, one; seven, two; and seven, five. In all four of these cases, 𝑎 minus 𝑏 is a prime number, where 𝑎 is an element of set 𝑋 and 𝑏 is an element of set 𝑌.