Set P has 4 elements and set Q has 5 elements. How many numbers of injections are defined from P to Q?
120
Understanding Injections from Set P to Set Q
The question asks us to find the number of injections that can be defined from set P to set Q. We are given the number of elements in set P and set Q.
What is an Injection?
An injection, also known as a one-to-one function, is a function from a set P to a set Q where each distinct element in P is mapped to a distinct element in Q. In simpler terms, no two different elements in P map to the same element in Q.
If \(f: P \to Q\) is an injection, then for any elements \(p_1, p_2 \in P\), if \(p_1 \neq p_2\), then \(f(p_1) \neq f(p_2)\).
Given Information
- Number of elements in set P, denoted as \(|P|\), is 4.
- Number of elements in set Q, denoted as \(|Q|\), is 5.
So, \(|P| = 4\) and \(|Q| = 5\).
Condition for Existence of Injections
An injection from set P to set Q can exist only if the number of elements in P is less than or equal to the number of elements in Q (\(|P| \le |Q|\)).
In this case, \(|P| = 4\) and \(|Q| = 5\). Since \(4 \le 5\), injections from P to Q are possible.
Calculating the Number of Injections
The number of injections from a set P with \(m\) elements to a set Q with \(n\) elements (where \(m \le n\)) is given by the number of permutations of selecting \(m\) elements from \(n\) elements. This is denoted by \(P(n, m)\) or \(_nP_m\) and calculated using the formula:
$$P(n, m) = \frac{n!}{(n-m)!}$$
Here, \(m = |P| = 4\) and \(n = |Q| = 5\). We need to calculate \(P(5, 4)\).
Applying the formula:
$$P(5, 4) = \frac{5!}{(5-4)!}$$
$$P(5, 4) = \frac{5!}{1!}$$
Now, let's calculate the factorials:
- \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
- \(1! = 1\)
Substitute these values back into the formula:
$$P(5, 4) = \frac{120}{1}$$
$$P(5, 4) = 120$$
Therefore, the number of injections from set P to set Q is 120.
| Concept | Formula / Definition | Application |
|---|---|---|
| Set P size, \(|P|\) | m | \(m = 4\) |
| Set Q size, \(|Q|\) | n | \(n = 5\) |
| Injections condition | \(m \le n\) | \(4 \le 5\) (Condition met) |
| Number of Injections | \(P(n, m) = \frac{n!}{(n-m)!}\) | \(P(5, 4) = \frac{5!}{(5-4)!}\) |
| Calculation | \(5! = 120\) | \(P(5, 4) = \frac{120}{1} = 120\) |
Revision Table: Functions Between Sets
| Type of Function | Condition on \(|P|\) (\(m\)) and \(|Q|\) (\(n\)) | Number of Functions |
|---|---|---|
| Any Function | No specific condition | \(n^m\) |
| Injection (One-to-one) | \(m \le n\) | \(P(n, m) = \frac{n!}{(n-m)!}\) |
| Surjection (Onto) | \(m \ge n\) | Given by inclusion-exclusion principle (more complex formula) |
| Bijection (One-to-one and Onto) | \(m = n\) | \(n!\) |
Additional Information: Types of Functions
Understanding different types of functions between sets is crucial in set theory and discrete mathematics.
- Injection (One-to-one function): As discussed, every element in the domain maps to a unique element in the codomain. If \(|P| > |Q|\), no injection exists.
- Surjection (Onto function): A function \(f: P \to Q\) is surjective if every element in the codomain Q is the image of at least one element in the domain P. This means the range of the function is equal to the codomain. If \(|P| < |Q|\), no surjection exists.
- Bijection (One-to-one correspondence): A function that is both an injection and a surjection. A bijection can only exist if \(|P| = |Q|\). If a bijection exists between two sets, they are said to have the same cardinality. The number of bijections between two sets of size n is \(n!\).
In this problem, we focused on injections from a set of size 4 to a set of size 5. Since \(4 \le 5\), injections are possible, and we calculated the number using the permutation formula.
Important Questions from Relations
If R and S are relatives from A to B, then which of the following statements are false?
(i) If R ≤ S, then \( S^{-1}\leq R^{-1} \)
(ii) If \(R \leq S,then \overline{S} \leq \overline{R}\)Let $g(.)$ is a function from $A$ to $B$, $f(.)$ is a function from $B$ to $C$, and their composition defined as $f(g(.))$ is a mapping from $A$ to $C$.
If $f(.)$ and $f(g(.))$ are onto (surjective) functions, which ONE of the following is TRUE about the function $g(.)$?
In the following question, select the related letters from the given alternatives.
MOB : ORG :: GRP : ?
The greatest integer function f : R → R given by f(x) = [x], (where [x] denotes the greatest integer), is _______
What is the scope of the definition of exponential function?
