I am lost, need to know how to do these types of problems.
How many one-to-one correspondences are there between the sets {a,b,c,d}and{1,2,3,4}if in each correspondence
A. a and c must correspond to even numbers
B. b must correspond to 3?
C. b must correspond to 3 and d to 4?
To solve these types of problems, we can follow a step-by-step approach:
Step 1: Count the number of elements in each set.
In this case, both sets have 4 elements.
Step 2: Analyze the conditions mentioned in the problem statement.
A. In each correspondence, a and c must correspond to even numbers.
B. b must correspond to 3.
C. b must correspond to 3 and d must correspond to 4.
Step 3: Identify the elements in the first set that satisfy the given conditions.
A. Since there are no even numbers in the set {a,b,c,d}, we can ignore this condition as it is not possible to satisfy it.
B. The condition is that b must correspond to 3.
C. The conditions are that b must correspond to 3 and d must correspond to 4.
Step 4: Count the number of ways to satisfy the given conditions.
B. There is only one element in the set {1,2,3,4} that is 3, so b must correspond to 3. The other elements can be paired with any of the remaining three elements.
Therefore, the number of one-to-one correspondences satisfying condition B is 3.
C. Similarly, there is one element that is 3 and one element that is 4. So, b must correspond to 3, and d must correspond to 4. The other elements can be paired with any of the remaining two elements.
Therefore, the number of one-to-one correspondences satisfying condition C is 2.
Step 5: Calculate the total number of one-to-one correspondences.
Since each condition is independent, we can find the total number of correspondences by multiplying the number of correspondences satisfying each condition.
For condition B, the number is 3.
For condition C, the number is 2.
So, the total number of one-to-one correspondences satisfying both conditions B and C is 3 * 2 = 6.
Therefore, the solution to each part of the problem is as follows:
A. No valid correspondences are possible.
B. There is exactly 1 correspondence possible.
C. There are exactly 6 correspondences possible.
By breaking down the problem into smaller steps and analyzing the conditions, we can determine the number of one-to-one correspondences between the two sets.