Introduction to Relations

Step 1: In a symmetric relation, every pair must have its reverse. Step 2: The reverse of ((3,5)) is ((5,3)). Step 3: For symmetry, interchange the first and second components.

Step 1: Transitivity requires ((a,c)) from ((a,b)) and ((b,c)). Step 2: From ((2,4)) and ((4,6)), the needed pair is ((2,6)). Step 3: Remove the common middle element and form the new pair.

Step 1: An empty relation has no ordered pair. Step 2: So it does not require pairs like ((a,a)). Step 3: In contrast, a reflexive relation requires all diagonal pairs.

Step 1: In an inverse relation, every pair is reversed. Step 2: ((1,2)) becomes ((2,1)), and ((2,1)) becomes ((1,2)). Step 3: So the inverse relation is the same relation.

Step 1: Symmetry requires ((2,1)) along with ((1,2)). Step 2: ((2,1)) is missing, so the relation is not symmetric. Step 3: Missing one reverse pair breaks symmetry.

Step 1: The domain is formed from first components. Step 2: The first components are (1,2,3). Step 3: Even in diagonal pairs, read the first component carefully.

Step 1: The range is formed from second components. Step 2: The second components are (1,2,3). Step 3: In an identity relation, domain and range can be the same.

Step 1: In (A\times B), the first component comes from (A) and the second from (B). Step 2: Hence the pairs are ((1,3)) and ((2,3)). Step 3: In Cartesian products, changing the order can change the answer.

Step 1: In (A\times B), the first component must come from (A). Step 2: In ((3,1)), the first component is (3), which is not in (A). Step 3: Position matters a lot in ordered pairs.

Step 1: Domain is formed from first components and range from second components. Step 2: The first components give (1,2), and the second components also give (1,2). Step 3: Even for a larger relation, find domain and range separately.

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.