Modern Mathematics. Made Simple by Patrick Murphy

By Patrick Murphy

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SWITCHING CIRCUITS A simple switch is represented by the type of diagram in Fig. 1, in which the switch is used to interrupt current flowing between Ρ and Q. The two-state on-off form of switch enables it to register the true-false nature of a statement. This allows us to refer to the statement and the switch by the same letter symbol. I ^ 1 Fig. 1 For the common language of switching circuits we use variables like x, y and ζ to represent both the statement and its corresponding switch. A false statement is indicated by an open switch (ofi") and the representative variable is given the value 0.

X: the rectangle has two equal sides y: the rectangle is a square 5. x: the triangle has two equal sides y: the triangle has two equal angles 6. Given that it is a positive integer prove that if N^ is odd then Η is odd. We conclude this chapter with a set of miscellaneous worked examples. Example 22. Consider the following proposition: If up is down the east is west, and if east is west, then ice is hot Therefore, if up is down, then ice is hot (i) (ii) Elementary Mathematical Logic 47 Construct a truth table and determine if the conclusion is true.

E. y-x. 9 shows that the theorem and its contraposidve theorem have the same set of truth values and the converse theorem has the same set of truth values as the inverse theorem. 9 It follows therefore that it may be possible to take advantage of this result if the contrapositive form of a theorem is easier to prove than the theorem itself, and we should consider the next example with this in mind. Example 21. Given that η is a positive integer prove that if Solution. Just to identify the foregoing remarks we suggest is odd then Η is odd.

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