A set is a well-defined collection, meaning you can clearly decide whether any object belongs to it or not. This clarity is essential for set operations and logic.
Which form lists elements directly?
A Set-builder form
B Roster form
C Function form
D Relation form
Roster form shows all elements explicitly inside braces, like {1,2,3}. It is best for small sets where listing every element is easy.
The empty set is written as
A {0}
B U
C { } with 1
D {}
The empty set has no elements and is written as {} or ∅. It is different from {0}, which contains one element, zero.
A set with one element is called
A Universal set
B Proper set
C Singleton set
D Power set
A singleton set contains exactly one element, like {5}. It is not empty, and it is useful when discussing subsets and power sets.
If A ⊆ B means
A A is subset of B
B A equals B only
C B is subset of A
D A is disjoint B
A ⊆ B means every element of A is also in B. A may be equal to B or smaller, depending on elements.
A ⊂ B indicates
A A equals B
B No common element
C Same cardinality
D Proper subset
A ⊂ B means A is a subset of B but not equal to B. So B has at least one element not in A.
Universal set is denoted by
A ∅
B P
C U
D ∩
The universal set U contains all objects under discussion in a problem. Other sets are subsets of U, depending on the defined context.
Union A ∪ B means
A Common elements only
B All elements of both
C Elements in neither
D Only elements of A
A ∪ B contains every element that is in A or in B or in both. It combines sets without repeating elements.
Intersection A ∩ B means
A All elements of both
B Elements in A not B
C Common elements only
D Elements in neither
A ∩ B includes only those elements that belong to both A and B. If no elements are common, the intersection is the empty set.
Difference A − B means
A In B not A
B Common elements
C All from A and B
D In A not B
A − B contains elements that are in A but not in B. It removes from A all elements that are also present in B.
Complement of A is
A U − A
B A ∩ U
C A − U
D U ∪ A
The complement A′ (or Aᶜ) contains all elements of the universal set U that are not in A. It depends on what U is.
If A and B are disjoint, then
A A ∩ B = U
B A ∪ B = ∅
C A − B = ∅
D A ∩ B = ∅
Disjoint sets have no common elements. Therefore their intersection is empty. Their union can still be non-empty if either set has elements.
De Morgan’s law says (A ∪ B)′ equals
A A′ ∪ B′
B A ∩ B
C A′ ∩ B′
D A′ − B′
De Morgan’s law converts complement of union into intersection of complements: (A ∪ B)′ = A′ ∩ B′. It helps simplify set expressions.
Another De Morgan’s law: (A ∩ B)′ equals
A A′ ∩ B′
B A′ ∪ B′
C A ∩ B′
D A′ − B′
Complement of an intersection becomes union of complements: (A ∩ B)′ = A′ ∪ B′. It is widely used in logic and probability.
Power set P(A) means
A All subsets of A
B All elements of A
C Only proper subsets
D Only singleton subsets
The power set P(A) is the set of all subsets of A, including ∅ and A itself. It is important for counting subsets.
If A has n elements, P(A) has
A n subsets
B 2n subsets
C 2^n subsets
D n^2 subsets
A set with n elements has 2^n subsets because each element has two choices: included or not included. This gives total combinations 2×2×…×2.
Cartesian product A × B contains
A Unordered pairs
B Only common elements
C Only subsets
D Ordered pairs
A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. Order matters, so (a,b) differs from (b,a).
If |A|=2 and |B|=3, then |A×B| is
A 5
B 9
C 6
D 23
The number of ordered pairs in A×B equals |A|·|B|. With 2 and 3 elements, total pairs are 2×3 = 6.
A relation from A to B is a subset of
A A × B
B A ∪ B
C A ∩ B
D P(A)
Any relation from A to B is defined as a set of ordered pairs (a,b) with a in A and b in B. Hence it is a subset of A×B.
Domain of a relation is
A Second elements
B All ordered pairs
C First elements
D Only repeated pairs
The domain is the set of all first components of ordered pairs in the relation. It tells which inputs from set A are actually related.
Range of a relation is
A First elements
B Codomain always
C Power set of B
D Second elements used
Range is the set of second components that actually appear in the relation. It may be smaller than codomain, which is the full target set B.
A relation on set A means
A Subset of A×B
B Subset of B×B
C Subset of A×A
D Any function only
A relation “on A” relates elements of A to elements of the same set, so it is a subset of A×A, like (a,a′) pairs.
Reflexive relation requires
A (a,b) for all pairs
B (a,a) for all a
C (a,b) implies (b,a)
D (a,b) implies (b,c)
A relation is reflexive if every element is related to itself. So for each a in A, the ordered pair (a,a) must be present.
Symmetric relation means
A (a,a) always present
B (a,b) ⇒ (a,c)
C Only one direction
D (a,b) ⇒ (b,a)
In a symmetric relation, if a is related to b, then b must be related to a. This property is common in “friendship” type relations.
Transitive relation means
A (a,b) ⇒ (b,a)
B (a,a) for all a
C (a,b) and (b,c) ⇒ (a,c)
D No pair repeats
Transitive means relations can “chain.” If a is related to b and b is related to c, then a must be related to c to satisfy transitivity.
An equivalence relation must be
A Reflexive, symmetric, transitive
B Only reflexive
C Only symmetric
D Only transitive
Equivalence relation groups elements into classes based on being “equivalent.” It must satisfy all three properties: reflexive, symmetric, and transitive.
Equivalence classes are always
A Overlapping always
B Empty always
C Only singleton
D Disjoint or same
For an equivalence relation, two equivalence classes are either identical or do not overlap. Together, all classes partition the set into clear groups.
Inverse relation R⁻¹ contains
A Same ordered pairs
B Only diagonal pairs
C Swapped ordered pairs
D Only symmetric pairs
If (a,b) is in R, then (b,a) is in R⁻¹. Inverse relation reverses the direction of every ordered pair.
Composition R∘S is defined when
A Range of S matches domain of R
B Both on same set only
C Both are functions only
D Sets must be equal always
For composition R∘S, you go from x via S to y, then via R from y to z. So the “middle” set must match properly.
A function is a special relation where
A One input, many outputs
B Each output has one input
C Only ordered pairs repeat
D Each input has one output
In a function, every element of the domain is assigned exactly one image in the codomain. This uniqueness of output defines a function.
Domain of a function means
A All outputs
B Only range
C Input set
D Only codomain
Domain is the set of all allowed input values for the function. The function must be defined for each element of the domain.
Codomain of a function is
A Actual outputs only
B Target set of outputs
C Domain again
D Only negative values
Codomain is the set in which outputs are expected to lie. Actual outputs form the range, which may be smaller than codomain.
A one-one function means
A Different inputs, same output
B All outputs used
C Different inputs, different outputs
D Constant output always
In a one-one (injective) function, no two distinct inputs share the same output. This property is important for having an inverse function.
Onto function means
A Domain equals codomain
B Many inputs one output
C Not defined everywhere
D Range equals codomain
A function is onto (surjective) if every element of the codomain is hit by at least one input. So the range covers the entire codomain.
Bijection means
A One-one and onto
B One-one only
C Onto only
D Into and many-one
A bijection is both injective and surjective, so it pairs each domain element with exactly one codomain element and covers the entire codomain.
Constant function has
A Different outputs always
B Output equals input
C Same output always
D No domain
In a constant function, every input maps to the same fixed value, like f(x)=5. It is usually not one-one unless domain has one element.
Identity function is
A f(x)=0
B f(x)=x²
C f(x)=|x|
D f(x)=x
Identity function maps each element to itself. It is one-one and onto when domain and codomain are the same set, and it acts like a neutral element in composition.
Many-one function means
A Same input, many outputs
B Range equals domain
C Different inputs, same output
D One-one always
Many-one function allows two or more different inputs to have the same output. Such functions cannot have a true inverse on the full domain.
Into function means
A Range equals codomain
B Range proper subset
C One-one always
D Constant always
Into function does not cover the entire codomain. Some elements of codomain are not images of any domain element, so range is a proper subset.
Modulus function is
A f(x)=x
B f(x)=1/x
C f(x)=⌊x⌋
D f(x)=|x|
Modulus function outputs the absolute value of x, making negative inputs positive. Its graph is symmetric about the y-axis and is not one-one on all real numbers.
Greatest integer function is
A ⌊x⌋
B ⌈x⌉
C |x|
D 1/x
The greatest integer function gives the largest integer less than or equal to x. It creates a step-like graph, constant on each interval [n, n+1).
A rational function is like
A Polynomial only
B |x| form
C p(x)/q(x)
D Step function only
Rational functions are ratios of two polynomials, p(x)/q(x), with q(x) not zero. They can have restrictions where the denominator becomes zero.
A polynomial function includes
A x in denominator
B Absolute value
C Floor brackets
D Non-negative powers
Polynomial functions have terms with non-negative integer exponents like x², x³, etc. They never have variables in denominators or fractional/negative powers.
Composition (f∘g)(x) means
A f(g(x))
B f(x)+g(x)
C g(f(x)) always
D f(x)g(x)
(f∘g)(x) means apply g first to x, then apply f to the result. Order matters, so f∘g is usually different from g∘f.
For (f∘g)(x) to exist
A g onto is needed
B f must be constant
C Range of g ⊆ domain of f
D g must be identity
Composition works only when outputs of g are valid inputs for f. So the range (or outputs) of g must lie inside the domain of f.
Inverse function exists when function is
A Many-one
B One-one and onto
C Constant
D Into only
A function must be bijective to have an inverse that is also a function. One-one ensures uniqueness, and onto ensures every codomain value is achieved.
If f has inverse, then
A f∘f = I
B f⁻¹∘f⁻¹ = I
C f+f⁻¹ = I
D f∘f⁻¹ = I
For invertible functions, composing f with its inverse returns the identity function on the appropriate set. Similarly, f⁻¹∘f also gives identity on the domain.
Inverse of composition is
A (f∘g)⁻¹ = f⁻¹∘g⁻¹
B (f∘g)⁻¹ = f∘g
C (f∘g)⁻¹ = g⁻¹∘f⁻¹
D No such rule
The inverse of a composition reverses order: first undo f, then undo g. So (f∘g)⁻¹ = g⁻¹∘f⁻¹, provided both inverses exist.
Graph of inverse is reflection about
A line y=x
B x-axis
C y-axis
D origin only
If a function has an inverse, their graphs are mirror images across the line y = x. This happens because input and output coordinates swap in inverse mapping.
A function is even if
A f(-x)=-f(x)
B f(x+T)=f(x)
C f(x)>0 always
D f(-x)=f(x)
Even functions satisfy f(-x)=f(x), showing symmetry about the y-axis. Common examples are x² and |x|. Odd functions satisfy f(-x)=-f(