A statement that is about every element of the domain is known as universal statement. It can also be defined as the negation of existential statement. Some examples are:
- n*n+n is an even number
- 4n-1 is a prime number
- 2 and 3 are only consecutive prime numbers
Let’s solve some sums based on this.
- Find a counter example for the statement 6n+1 is a prime.
Let us consider a domain of positive integers. If we take value of n=1, then the result is 7 i.e. prime. For n=2, the result is 13 i.e. prime and so on. But when we take n=4, the result is 25, which is not a prime.
So, we conclude that the counter example for 6n+1 is n=4.
2. Is the statement if n is a multiple of 3 then n is a multiple of 6 true or false?
Let the domain be positive integers. For n to be a multiple of 3, n=3x where x is a positive integer. Similarly, for n to be a multiple of 6, n=6x. When we take x=3, n=3x i.e. n=9. 9 is a multiple of 3 but not a multiple of 6.
So, we conclude that the given statement is false for x=3.