Relations and Functions — Chapter Test

A binary operation is a function from A × A to A. |A × A| = n². Number of functions = n^(n²).

Using the inclusion-exclusion principle: 3⁵ − ³C₁(2⁵) + ³C₂(1⁵) = 243 − 3(32) + 3(1) = 243 − 96 + 3 = 150.

Division of integers is not always an integer (e.g., 3 ÷ 2 = 1.5 ∉ Z). Addition, subtraction, and multiplication are closed on Z.

The number of one-one functions from a set of size m to a set of size n (where m ≤ n) is ⁿPₘ. Here it is ⁴P₃ = 4 × 3 × 2 = 24.

f(f(x)) = a(ax+b)+b = a²x + ab + b. Equating coefficients: a² = 16 → a = 4 or −4. If a = 4, 4b+b = −15 → 5b = −15 → b = −3. Sum = 1. If a = −4, −4b+b = −15 → −3b = −15 → b = 5. Sum = 1.

If there is a direct train from station x to y, the same train runs from y to x, making it symmetric. It isn't necessarily reflexive (a train from a station to itself is trivial) or transitive.

A person cannot earn more than themselves (not reflexive). If A earns more than B, B cannot earn more than A (not symmetric). If A earns more than B, and B more than C, A earns more than C (transitive).

For a given x, there could be multiple y satisfying the equation, or none. The relation is not a function because one x may correspond to more than one y (or no y). For example, x=5 gives 2y=70 → y=35; x=15 gives 2y=40 → y=20. But each x gives exactly one y actually: y = (85−3x)/2. Since it's a relation from N to N, it associates exactly one y to each x where 85−3x is even and positive. This defines a function from a subset of N. Since y = (85−3x)/2 gives a unique y for each x, it is a function. However, the domain is not all N, only those x where (85−3x) is a positive even integer. But the question says relation 'from N to N'. A function can have domain a proper subset. Actually, a relation R from A to B is a function if for every a ∈ A there is at most one b such that (a,b) ∈ R. Here, for each x ∈ N, there is at most one y. So it is a function (partial function). But the question might intend that not all x give a natural y, so it's not a function 'from N to N'? Usually 'function from A to B' requires domain = A. Since some x ∈ N do not yield y ∈ N, it's not a function from N to N. We go with 'not a function' because domain is not all of N. Yes, a function must be defined on the whole domain. So answer is 'not a function'.

f(1.2) = 1 and f(1.8) = 1, so it is many-one. The range of f is Z (set of integers), which is not equal to the co-domain R. Thus, it is not onto.

Every number divides itself, so R is reflexive. If a divides b and b divides c, then a divides c, so R is transitive. However, 2 divides 4 but 4 does not divide 2, so it is not symmetric.

a * b = a/b, but b * a = b/a, so not commutative. Also, (a * b) * c = (a/b)/c = a/(bc), whereas a * (b * c) = a/(b/c) = (ac)/b. Hence, not associative.

On [0, ∞), f(x₁) = f(x₂) → x₁² = x₂² → x₁ = x₂ (since non-negative) → one-one. Range of f is [0, ∞) → onto. Hence bijective.

2ˣ is strictly increasing → one-one. Range is (0, ∞), not all real numbers → not onto.

Commutative: (a+b)/2 = (b+a)/2. Associative: (a∗b)∗c = ((a+b)/2 + c)/2 = (a+b+2c)/4. a∗(b∗c) = (a + (b+c)/2)/2 = (2a+b+c)/4. Not equal in general, so not associative.

A line cannot be perpendicular to itself (not reflexive). If L₁ ⊥ L₂, then L₂ ⊥ L₁ (symmetric). If L₁ ⊥ L₂ and L₂ ⊥ L₃, then L₁ is parallel to L₃, not perpendicular (not transitive).