Introduction to Relations

Correct answer: A. Universal relation

Step 1: A relation containing all pairs of (A\times A) is called the universal relation. Step 2: No possible ordered pair is left out. Step 3: If all pairs are present, identify it as universal.

Correct answer: A. Yes

Step 1: Symmetry needs the reverse of every non-identical pair. Step 2: ((1,2)) has ((2,1)), and self-pairs reverse to themselves. Step 3: Pairs of the form ((a,a)) do not create a problem for symmetry.

Correct answer: A. Because ((3,2)) is missing

Step 1: Symmetry requires the reverse of every pair. Step 2: The reverse of ((2,3)) is ((3,2)), which is not given. Step 3: Having some reverse pairs is not enough; all are required.

Correct answer: A. ((1,3))

Step 1: Transitivity needs ((a,c)) from ((a,b)) and ((b,c)). Step 2: From ((1,2)) and ((2,3)), we get ((1,3)). Step 3: For transitivity, take the first element of the first pair and the second element of the second pair.

Correct answer: A. No

Step 1: The reverse of ((1,2)), namely ((2,1)), should be present. Step 2: Since ((2,1)) is missing, the relation is not symmetric. Step 3: Do not get confused by self-pairs; check reverse non-self pairs.

Correct answer: A. ((4,4))

Step 1: In an identity relation, every element is related to itself. Step 2: Since (4) is an element of the set, ((4,4)) must be present. Step 3: Identity relation does not include pairs of different elements.

Correct answer: A. (R\subseteq A\times A)

Step 1: A relation on (A) means both positions in the ordered pairs come from (A). Step 2: Therefore, all pairs come from (A\times A). Step 3: Understand the difference between a relation on (A) and a relation from (A) to (B).

Correct answer: B. 6

Step 1: In (A\times B), there are (2) choices for the first position and (3) choices for the second. Step 2: Total pairs are (2\times3=6). Step 3: For different sets also, multiply to count Cartesian product pairs.

Correct answer: A. Reflexive, symmetric and transitive

Step 1: All self-pairs are present, so the relation is reflexive. Step 2: The reverse of each self-pair is itself, and chains also remain valid. Step 3: Identity relation is a common example of a reflexive, symmetric, and transitive relation.

Correct answer: A. Equivalence relation

Step 1: An equivalence relation needs three properties. Step 2: These properties are reflexivity, symmetry, and transitivity. Step 3: To identify an equivalence relation, check all three conditions separately.

Correct answer: A. No

Step 1: An equivalence relation requires all three properties together. Step 2: Since transitivity is absent, the condition is not complete. Step 3: Missing even one property prevents a relation from being an equivalence relation.

Correct answer: A. ((3,1))

Step 1: In a relation on (A), both entries must come from (A). Step 2: In ((3,1)), the element (3) is not in (A). Step 3: While checking a pair, first see whether its entries belong to the given set.

Correct answer: A. ({(x,x),(y,y)})

Step 1: In an identity relation, each element is related only to itself. Step 2: For (x), we need ((x,x)), and for (y), we need ((y,y)). Step 3: Pairs with different elements are not part of the identity relation.

Correct answer: A. ((1,2))

Step 1: Transitivity needs ((a,c)) from ((a,b)) and ((b,c)). Step 2: Here ((1,2)) and ((2,2)) again require ((1,2)). Step 3: Sometimes the required pair is already one of the given pairs.

Correct answer: A. Identity relation

Step 1: Self-pairs are of the form ((a,a)). Step 2: A relation with only such pairs is the identity relation when every element has its self-pair. Step 3: Identify identity relation by the absence of pairs with different elements.

Correct answer: A. (\varnothing)

Step 1: An empty relation contains no ordered pair. Step 2: (\varnothing) denotes the empty set. Step 3: If even one pair is present, the relation is not empty.

Correct answer: A. ({(1,1),(1,2),(2,1),(2,2)})

Step 1: A universal relation contains all pairs of (A\times A). Step 2: For (A={1,2}), four pairs are possible, and the first option has all four. Step 3: Match both the count and the pairs before choosing.

Correct answer: A. Yes

Step 1: It has all self-pairs, so it is reflexive. Step 2: Every pair has its reverse, and all possible pairs are included, so transitivity also holds. Step 3: A universal relation on a set can be an example of an equivalence relation.

Correct answer: A. Because it is not symmetric

Step 1: Symmetry is necessary for an equivalence relation. Step 2: ((1,2)) is present but ((2,1)) is missing, so symmetry fails. Step 3: Do not forget to check all three properties for equivalence relation.

Correct answer: A. Universal relation

Step 1: If a relation is exactly equal to (A\times A), it contains all pairs. Step 2: A relation containing all pairs is universal. Step 3: When (R=A\times A) is given, identify it as universal immediately.

Correct answer: A. Relation on (A)

Step 1: A relation on (A) is any subset of (A\times A). Step 2: Therefore, if (R\subseteq A\times A), then (R) is a relation on (A). Step 3: Always think of a relation as a set of ordered pairs.

Correct answer: A. ((2,2))

Step 1: In an identity relation, every element is related to itself. Step 2: Since (2) is in the set, ((2,2)) will be included. Step 3: Pairs with different elements are not included in the identity relation.

Correct answer: C. 25

Step 1: In (A\times A), elements of (A) appear in both positions. Step 2: Therefore, total pairs are (5\times5=25). Step 3: Use (n^2) for counting pairs in (A\times A).

Correct answer: A. Reflexive relation

Step 1: In a reflexive relation, every element must have its self-pair. Step 2: If any element is left out, reflexivity is not complete. Step 3: Remember reflexivity as self-relation for all elements.

Correct answer: A. (x)

The function adds 4 to its input. To find the requested difference, first calculate the function's value at zero. Subtracting this fixed value from the general value removes the added constant, so only the input-related part remains. This illustrates that a vertical shift by a constant disappears when two function values are compared in this way.

Substitute zero for the input: \(f(0)=0+4=4\). The general value is \(f(x)=x+4\). Therefore, \(f(x)-f(0)=(x+4)-4=x\). The constant terms cancel, leaving \(x\), which is option A. Option B is just the original function, and option D would incorrectly ignore the variable term.