Conditional Statements Theorems are usually given in the form of conditional statements. Look at conditional statements in general. They all have the form: If P, then Q where P is called the hypothesis and Q is the conclusion. Example: If it snows tomorrow, there will be no class. P = it snows tomorrow. Q = there is no class. The contrapositive of the statement is: If not Q, then not P. If there is class tomorrow, then it will not be snowing. The contrapositive of a true statement is true. The converse of the statement is: If Q then P: If there is no class tomorrow, then it will snow. NOT A TRUE STATEMENT. In general, the converse of a true statement is not true. The inverse of the statement is: If not P, then not Q. If it does not snow tomorrow, then there will be class. False. In general, the inverse of a true statement is not true. Example: If a rectangle is equilateral, then it is a square. Contrapositive: If a rectangle is not a square, then it is not equilateral. (true) Converse: If a rectangle is a square, then it is equilateral. (true) Inverse: If a rectangle is not equilateral, then it is not a square. (True) Theorem 1.7: An indexed set S v1 , v 2 ,..., v p of two or more vectors is linearly dependent iff at least one of the vectors in S is a linear combo of the others. In fact, if S is linearly dependent and v1 ≠ 0, then some vector vj (j ≥ 2) is a linear combo of the preceding vectors. The statement: If an indexed set S v1 , v 2 ,..., v p of two or more vectors is linearly dependent, then at least one of the vectors in S is a linear combo of the others. (True) The contrapositive: If none of the vectors in S is a linear combo of the others, the indexed set S v1 , v 2 ,..., v p of two or more vectors is linearly independent. (True) Since Theorem 1.7 says “if and only if”, the converse and the inverse are also true. Converse: If at least one of the vectors in S is a linear combination of the others, the indexed set S v1 , v 2 ,..., v p of two or more vectors is linearly dependent. (True) Inverse: If an indexed set S v1 , v 2 ,..., v p of two or more vectors is linearly independent, then none of the vectors in S is a linear combination of the others. Theorem 1.8: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent (i.e. any set v1 , v 2 ,, v p is linearly dependent if p > n). Only the contrapositive is true: If a set is linearly independent, then the set contains no more vectors than there are entries in each vector. Theorem 9: If a set of vectors S v1 , v 2 ,, v p contains the zero vector, it is linearly dependent. Only the contrapositive is true: If a set of vectors S v1 , v 2 ,, v p is linearly independent, the set does not contain the zero vector. Negation of statements In general, to negate a statement you insert the word not. Ex: We are in building 28. Negation: We are not in building 28. Ex: The sky is not green. Neg: The sky is green. It gets more complicated when you are making a general statement about a group of things. Ex: Men are taller than women. Means: all men are taller than all women. All it takes to negate this is one exception: There is at least one man who is shorter than at least one woman. ►To negate a statement using “all”, use “there exists one” that does not meet the requirements. Ex: No dog can sing. Or All dogs do not sing. Neg: there is at least one dog that can sing. Ex: Some birds can swim. Or there is at least one bird that can swim. Neg: There is not at least one bird that can swim. Or No birds can swim. Note: a statement and its negation cannot both be true. Converse, Inverse and Contrapositive If it snows tomorrow, there will be no class. P = it snows tomorrow. Q = there is no class. The converse of the statement is: If Q then P: If there is no class tomorrow, then it will snow. The inverse of the statement is: If P, then not Q. If it snows tomorrow, then there will be class. The contrapositive of the statement is: If not Q, then not P. If there is class tomorrow, then it will not be snowing.