What are 4 inductive reasoning examples and 4 deductive reasoning examples?
Inductive reasoning can be thought of as reasoning from examples. Examples include:
1. Assuming that the sum of two even numbers is even because the sum is even for the 100 examples you tried.
2. Assuming that the base angles of an isosceles triangle are congruent since you drew 20 such triangles and measured the angles.
3. Finding the next number in a given sequence. Note that there are an infinite number of sequences that start with any given finite set of numbers. For instance, given the sequence 1,2,4,8,16,... you might infer the next number to be 32; but the sequence I have in mind has the next number 31.
4. Goldbach's conjecture: the modern version states that every even number greater than 4 can be written as the sum of two primes. This is still unproven, but known to be true for huge numbers. See reference.
Deductive reasoning starts from a given set of facts and proceeds logically to a conclusion. If the underlying facts are correct and there is no flaw in the reasoning, the result will be correct for all time. Examples include:
1. Prove that the sum of 2 even numbers is even. The two numbers can be written as 2j and 2k, where j,k are integers. Then 2j+2k=2(j+k) which is even. Note the difference between this and number 1 above.
2. Prove the base angles of an isosceles triangle are congruent. A computer proved this by showing triangle ABC is congruent to the reflection triangle ACB, and thus the corresponding angles are congruent. Again, note the difference with number 2 above.
3. Given the sequence of triangular numbers 1,3,6,10,15,... prove that the nth number is (1/2)n(n+1). See reference. Note that this is a specific sequence as opposed to number 3 above.
4. Prove all right angles are congruent. The measure of a right angle is 90 by definition, so all right angles have the same measure and thus are congruent.