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.

Revision Table: Functions Between Sets

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.