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If a statement and its converse are both true, then the conditions in the hypothesis of the statement are necessary and sufficient for its conclusion.

For example, the statement if angles are right angles, then they are congruent and supplementary is true, and its converse, if angles are congruent and supplementary, then they are right angles is also true. Hence, being right angles is necessary and sufficient for the angles to be congruent and supplementary.



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If a statement is true and its converse is false, then the conditions in the hypothesis of the statement are sufficient but not necessary for its conclusion.

Fig. 6-7

The statement if angles are right angles, then they are congruent is true, and its converse, if angles are congruent, then they are right angles, is false. Hence, being right angles is sufficient for the angles to be congruent. However, the angles need not be right angles to be congruent.

Costing Inventory: First-In, First-Out (FIFO)

If a statement is false and its converse is true, then the conditions in the hypothesis are necessary but not sufficient for its conclusion.





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The statement if angles are supplementary, then they are right angles is false, and its converse, if angles are right angles, then they are supplementary, is true. Hence, angles need to be supplementary to be right angles, but being supplementary is not sufficient for angles to be right angles.

PRINCIPLE 4:

If a statement and its converse are both false, then the conditions in the hypothesis are neither necessary nor sufficient for its conclusion.

Thus the statement if angles are supplementary, then they are congruent is false, and its converse, if angles are congruent, then they are supplementary, is false. Hence, being supplementary is neither necessary nor sufficient for the angles to be congruent.

When the Conditions in the Hypothesis of a Statement are Necessary or Sufficient to Justify its Conclusion Principle 1 2 3 4 Statement True True False False Converse True False True False Sufficient Yes Yes No No Necessary Yes No Yes No

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Determining necessary and sufficient conditions For each of the following statements, determine whether the conditions in the hypothesis are necessary or sufficient to justify the conclusion. (a) A regular polygon is equilateral and equiangular. (b) An equiangular polygon is regular. (c) A regular polygon is equilateral. (d) An equilateral polygon is equiangular.

goods that were bought rst ( rst-in) are the rst goods to be sold ( rst out), and the goods that remain on hand (ending inventory) are assumed to be made up of the latest costs. Therefore, for income determination, earlier costs are matched with revenue and the most recent costs are used for balance sheet valuation. This method is consistent with the actual ow of costs, since mer chandisers attempt to sell their old stock rst. (Perishable items and highfashion items are examples.) FIFO is the most widely used inventory method of those that will be discussed. Under FIFO, those goods left at the end of the period are considered to be those received last. Therefore, the 320 units on hand on December 31 would be costed as follows: Most recent purchase (10/4) Next most recent purchase (6/6) Ending inventory 250 units @ $10 = $2,500 70 units @ $ 9 = 630 320 units $3,130

(a) Since the statement and its converse are both true, the conditions are necessary and sufficient. (b) Since the statement is false and its converse is true, the conditions are necessary but not sufficient. (c) Since the statement is true and its converse is false, the conditions are sufficient but not necessary. (d) Since both the statement and its converse are false, the conditions are neither necessary nor sufficient.

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