Types of relations

Correct answer: A. because ((1,1)) is absent

Step 1: ((1,2)) and ((2,1)) are present. Step 2: Transitivity requires ((1,1)). Step 3: Since ((1,1)) is absent, the relation is not transitive.

Correct answer: A. ({1,2}) and ({3})

Step 1: Elements (1) and (2) are related in both directions, so they are in the same class. Step 2: Element (3) is related only to itself. Step 3: Therefore the classes are ({1,2}) and ({3}).

Correct answer: A. equivalence relation

Step 1: Every integer has the same parity as itself, so reflexivity holds. Step 2: The same-parity condition remains true when reversed. Step 3: Same parity also passes through a middle element, so it is an equivalence relation.

Correct answer: A. equivalence relation

Step 1: (a-a=0) is divisible by (5), so the relation is reflexive. Step 2: If (a-b) is divisible by (5), then (b-a) is also divisible by (5). Step 3: The sum of two such differences is also divisible by (5), so it is an equivalence relation.

Correct answer: A. because (1\le2) but (2\le1) is false

Step 1: Symmetry requires the reverse relation to also be true. Step 2: (1\le2) is true, but (2\le1) is false. Step 3: Therefore the (\le) relation is not symmetric.

Correct answer: A. because (a<a) is never true

Step 1: Reflexivity requires (aRa) for every (a). Step 2: Here (aRa) means (a<a). Step 3: Since (a<a) is never true, the relation is not reflexive.

Correct answer: A. transitivity

Step 1: If (a<b) and (b<c), then (a<c). Step 2: This is exactly the condition for transitivity. Step 3: Hence the less-than relation is transitive.

Correct answer: A. identity relation

Step 1: The condition (a=b) gives only pairs of equal elements. Step 2: Such pairs are of the form ((a,a)). Step 3: Therefore it is called the identity relation.

Correct answer: A. symmetry

Step 1: If (a\ne b), then (b\ne a) is also true. Step 2: Thus the reverse of every related pair is also related. Step 3: Hence the relation is symmetric.

Correct answer: A. no

Step 1: Reflexivity requires (aRa). Step 2: Here (aRa) means (a\ne a), which is false. Step 3: Therefore the relation is not reflexive.

Correct answer: A. because ((1,3)) is absent

Step 1: The relation appears reflexive and symmetric. Step 2: But ((1,2)) and ((2,3)) require ((1,3)). Step 3: Since ((1,3)) is absent, transitivity fails and it is not an equivalence relation.

Correct answer: A. yes

Step 1: All diagonal pairs are present, so it is reflexive. Step 2: ((1,2)) is present but ((2,1)) is absent, so antisymmetry is not violated. Step 3: No required transitive pair is missing, so it is a partial order relation.

Correct answer: A. no

Step 1: The universal relation contains both ((1,2)) and ((2,1)). Step 2: Since (1\ne2), having both directions violates antisymmetry. Step 3: Therefore it is not antisymmetric.

Correct answer: A. yes

Step 1: The identity relation contains only pairs of the form ((a,a)). Step 2: It has no two-way pairs between distinct elements. Step 3: Therefore the identity relation is antisymmetric.

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

Step 1: Symmetry requires the reverse of every pair. Step 2: The reverse of ((1,2)) is ((2,1)). Step 3: Thus ((2,1)) must be added; ((3,2)) would also be needed for complete symmetry.

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

Step 1: ((1,2)) and ((2,3)) form a chain. Step 2: Transitivity requires ((1,3)). Step 3: Therefore ((1,3)) must be added.

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

Step 1: To make the relation reflexive, all diagonal pairs are needed. Step 2: These are ((1,1)), ((2,2)), and ((3,3)). Step 3: Among the given options, ((1,1)) is a necessary diagonal pair.

Correct answer: A. ({(1,1),(2,2),(3,3)})

Step 1: A reflexive relation must contain all diagonal pairs. Step 2: The smallest such relation contains only these compulsory pairs. Step 3: Therefore the smallest reflexive relation is the identity relation.

Correct answer: A. universal relation

Step 1: All possible pairs on a set are in (A\times A). Step 2: The relation containing all these pairs is the universal relation. Step 3: Hence it is considered the largest relation.

Correct answer: A. empty relation

Step 1: A relation with no pair is called the empty relation. Step 2: It contains no ordered pair at all. Step 3: Hence it is considered the smallest relation.

Correct answer: A. symmetric relation

Step 1: A symmetric relation must contain the reverse of every pair. Step 2: ((1,2)) is present, but ((2,1)) is absent. Step 3: Therefore the relation definitely cannot be symmetric.

Correct answer: A. transitivity

Step 1: All diagonal pairs give reflexivity. Step 2: Having the reverse of every pair gives symmetry. Step 3: To prove equivalence, transitivity still needs to be checked.

Correct answer: A. 10

Composition means evaluate the inner function first and then apply the outer function. Here \((g\circ f)(1)=g(f(1))\). Compute \(f(1)=1+4=5\), then \(g(5)=2\cdot5=10\). Therefore the correct answer is 10. A common mistake is to evaluate \(g(1)\) first or to miscalculate \(f(1)\), which could lead to 9 or 8 as wrong choices. Exam tip: always perform the inner function evaluation first and keep track of the order of composition.

Correct answer: A. 11

Step 1: \(f(1)=1^2+2\cdot1=1+2=3\). Step 2: \(f(2)=2^2+2\cdot2=4+4=8\). So the sum is \(3+8=11\). The closest distractor C (10) could arise if someone mis-evaluates \(f(1)\) as 2, giving \(2+8=10\); that is incorrect because \(f(1)=3\). Exam tip: evaluate the function at each input separately (compute powers and multiplications first) and then add the results.

Correct answer: A. 4

Core idea: \((f-g)(x)=f(x)-g(x)\). Evaluate each function at \(x=2\): \(f(2)=4\times2=8\) and \(g(2)=2+2=4\). Then subtract: \((f-g)(2)=8-4=4\). Option C (8) is a common mistake — it is just \(f(2)\) and ignores subtracting \(g(2)\). Option D (2) equals \(f(2)/g(2)=8/4\), a division confusion, so it is incorrect here. Exam tip: Compute individual function values first at the given x, then apply the indicated operation (sum, difference, product, or quotient).