Introduction to Relations

Step 1: A reflexive relation must contain each element paired with itself. Step 2: Here ((1,1),(2,2),(3,3)) are all present so the relation is reflexive. Step 3: Extra pairs do not break reflexivity.

Step 1: Reflexivity needs the diagonal pair for every element. Step 2: For element (3), ((3,3)) is missing so the condition fails. Step 3: For reflexivity check every element of the base set.

Step 1: In a symmetric relation every pair must have its reverse pair. Step 2: The reverse of ((1,2)) is ((2,1)). Step 3: To check symmetry interchange the first and second components.

Step 1: For ((2,3)), symmetry requires ((3,2)). Step 2: The pair ((3,2)) is not present in the relation. Step 3: If even one reverse pair is missing, the relation is not symmetric.

Step 1: Transitivity requires ((a,c)) from ((a,b)) and ((b,c)). Step 2: From ((1,2)) and ((2,3)), we need ((1,3)). Step 3: Remove the common middle element and join the first and last elements.

Step 1: In ((1,2)) and ((2,3)), the middle element (2) matches. Step 2: Transitivity requires ((1,3)), but it is missing. Step 3: While checking transitivity focus on connected pairs.

Step 1: The smallest reflexive relation contains only necessary diagonal pairs. Step 2: For four elements the pairs are ((1,1),(2,2),(3,3),(4,4)). Step 3: When the word smallest appears do not add extra pairs.

Step 1: The smallest reflexive relation has one diagonal pair for each element. Step 2: For five elements there are five such pairs. Step 3: The number of pairs in the smallest reflexive relation equals the number of elements.

Step 1: The universal relation is equal to (A\times A). Step 2: For (3) elements, (3^2=9) pairs are formed. Step 3: A universal relation includes all possible ordered pairs.

Step 1: In the identity relation each element is related only to itself. Step 2: For (a,b,c), the pairs are ((a,a),(b,b),(c,c)). Step 3: The number of identity pairs equals the number of elements.

Step 1: The number of pairs in (A\times B) is (n(A)n(B)). Step 2: Here (2\times3=6) pairs are formed. Step 3: For different sets multiply the numbers of elements.

Step 1: In (B\times A), the first component comes from (B) and the second from (A). Step 2: In ((3,1)), (3) is from (B) and (1) is from (A). Step 3: Always read the order carefully in Cartesian products.

Step 1: The domain is formed from the first components of ordered pairs. Step 2: The first components are (1,2,2), so the domain is ({1,2}). Step 3: Write repeated elements only once in a set.

Step 1: The range is formed from the second components of ordered pairs. Step 2: The second components are (3,3,4), so the range is ({3,4}). Step 3: Remember the different positions for domain and range.

Step 1: The domain contains first components. Step 2: The first components of the relation are (a,b,c), so (b) is in the domain. Step 3: Letters and numbers can both occur but position matters most.

Step 1: The range contains second components. Step 2: Here the second components are (2,3,2), so (2) belongs to the range. Step 3: To find the range look at the right-side components.

Step 1: In an inverse relation the two components of every pair are reversed. Step 2: ((1,4),(2,5),(3,6)) become ((4,1),(5,2),(6,3)). Step 3: While finding inverse do not miss any pair.

Step 1: To get (R), reverse every pair of (R^{-1}). Step 2: ((2,1)) gives ((1,2)) and ((3,2)) gives ((2,3)). Step 3: The inverse of the inverse is the original relation.

Step 1: (R=R^{-1}) means every pair is present with its reverse. Step 2: This is the main idea of a symmetric relation. Step 3: Symmetry can also be recognized through the inverse relation.

Step 1: Diagonal pairs reverse to themselves. Step 2: The non-diagonal pair ((1,2)) has its reverse ((2,1)) present. Step 3: In symmetry check the reverse of every non-diagonal pair.

Step 1: Every element is related to itself. Step 2: There is no pair with different components so it is the identity relation. Step 3: Identify the identity relation through diagonal pairs.

Step 1: (A\times A) contains all possible ordered pairs. Step 2: A relation containing all these pairs is called the universal relation. Step 3: When you see the full Cartesian product think of universal relation.

Step 1: The domain is formed from first components. Step 2: An empty relation has no pair so there is no first component. Step 3: The domain and range of an empty relation are both empty.

Step 1: The range is formed from second components. Step 2: An empty relation has no ordered pair so it has no second component. Step 3: In an empty relation both domain and range are empty.

Step 1: (A\times A) has (2^2=4) pairs. Step 2: A set with four elements has (2^4=16) subsets. Step 3: The total number of relations equals the number of subsets.