Connecting words/ phrases – consolidating the understanding of “if and only if (necessary and sufficient) condition”, “implies”, “and/or”, “implied by”, “and”, “or”, “there exists” and their use through variety of examples related to real life and Mathematics
Connecting words are used to combine two or more statements. In this case, each statement is denoted as a component statement of a compound statement.
For example,all rational numbers are real numbers and all real numbers are complex numbers. In this sentence, there are two component statements connected with ‘and’.
- All rational numbers are real numbers.
- All real numbers are complex numbers.
Connectives: And, Or
And rule: When all the component statements connected with and are true then the compound statement is true. If at least one or more component statements connected with and are false then the compound statement is false.
Example: 0 is less than every positive integer and every negative integer.
The second statement is false, so the compound statement is false.
Example: All living things have two hands and two legs
There are two statements in this sentence:
- All living things have two hands
- All living things have two eyes
This is not a compound statement, here and refers to two separate entities: hands and legs.
When one or all component statements of a compound statement are true then the compound statement is also true. It is false when all the component statements are false.
Example: Delhi is the capital of Haryana or Rajasthan
Since both the statements are false, the compound statement is false.
Example: Two lines intersect at a point or they are parallel
When the first statement is true, the second is false and vice versa. Therefore, the compound statement is true.
Quantifiers: “ there exists”, ‘for all’, ‘for every’
Quantifiers are used to show the dependence of one statement over others.
Example: For every prime number x, is an irrational number.
Here, assume that S denotes a set all the prime numbers, then for every x in the set S, is an irrational number.
For every, or for all can be interpreted as all the quantities satisfy a specific property.
- For every positive number a there exists a positive number why such that b>a
- There exists a positive number b such that for every positive number a, we have b>a
Even though these sentences may look similar but they’re not the same thing. The first statement shown above is true and the second statement is false. For a piece of mathematical writing to make sense, all of the symbols must be carefully introduced and each symbol must be introduced precisely at the right place and not before not after.
Implications: “implies”, implied by’
X implies Y is written as. The symbol stands for implies which means that x is true then y must also be true.
Contrapositive and converse: ‘if-then’, ‘only if’ and if and only if’
These statements are true because if one is true then the other must also be true. For contrapositive statement show that ~q is true, then ~p is true.
If and only if or iff is represented by a symbol is meant for equivalence of the two statements where they are necessary and sufficient for each other. Consider the example where a and b are both integers.If one of a and b is odd and the other is even then (a+b) is odd can be written as:
One of a and b is odd and the other is even (a+b) is odd
In this case, it goes both ways because if (a+b) is odd and a and b are both integers, then one of them must be odd and the other even.