‘And’ is false if any of its component statements is false (this includes the case that some of its component statements are false or all of its component statements are false). Example Write the component statements of the following compound statements and check whether the compound statement is true or false. A line is straight and extends indefinitely in both directions. is less than every positive integer and every negative integer.
(iii) All living things have two legs and two eyes. Solution (i) The component statements are p: A line is straight. q: A line extends indefinitely in both directions. MATHEMATICAL REASONING Both these statements are true, therefore, the compound statement is true.
The component statements are p: is less than every positive integer. q: is less than every negative integer. The second statement is false. Therefore, the compound statement is false.
(iii) The two component statements are p: All living things have two legs. q: All living things have two eyes. Both these statements are false. Therefore, the compound statement is false.
Now, consider the following statement. p: A mixture of alcohol and water can be separated by chemical methods. This sentence cannot be considered as a compound statement with “And”. Here the word “And” refers to two things – alcohol and water.
This leads us to an important note.